direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C23.36C23, (C4×D4)⋊8C14, (C4×Q8)⋊7C14, (D4×C28)⋊37C2, (Q8×C28)⋊27C2, (C2×C42)⋊10C14, C22⋊Q8⋊22C14, C42⋊C2⋊9C14, C4⋊D4.10C14, C4.4D4⋊17C14, C42.34(C2×C14), C42.C2⋊13C14, C42⋊2C2⋊10C14, C28.274(C4○D4), (C2×C14).349C24, (C4×C28).373C22, (C2×C28).960C23, C23.7(C22×C14), (D4×C14).318C22, C22.D4⋊16C14, (C22×C14).87C23, C22.23(C23×C14), (Q8×C14).266C22, (C22×C28).511C22, (C2×C4×C28)⋊23C2, C4.56(C7×C4○D4), C4⋊C4.64(C2×C14), C2.12(C14×C4○D4), (C7×C22⋊Q8)⋊49C2, C22.3(C7×C4○D4), (C2×D4).63(C2×C14), C14.231(C2×C4○D4), (C7×C4.4D4)⋊37C2, (C7×C4⋊D4).20C2, (C2×Q8).53(C2×C14), (C7×C42.C2)⋊30C2, (C7×C42⋊2C2)⋊21C2, (C7×C42⋊C2)⋊30C2, (C2×C14).51(C4○D4), (C7×C4⋊C4).387C22, C22⋊C4.12(C2×C14), (C22×C4).54(C2×C14), (C2×C4).17(C22×C14), (C7×C22.D4)⋊35C2, (C7×C22⋊C4).146C22, SmallGroup(448,1312)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 322 in 234 conjugacy classes, 154 normal (62 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×2], C22 [×8], C7, C2×C4 [×6], C2×C4 [×6], C2×C4 [×10], D4 [×6], Q8 [×2], C23, C23 [×2], C14 [×3], C14 [×4], C42 [×4], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×8], C22×C4 [×3], C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C28 [×4], C28 [×10], C2×C14, C2×C14 [×2], C2×C14 [×8], C2×C42, C42⋊C2 [×2], C4×D4, C4×D4 [×2], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C42⋊2C2 [×2], C2×C28 [×6], C2×C28 [×6], C2×C28 [×10], C7×D4 [×6], C7×Q8 [×2], C22×C14, C22×C14 [×2], C23.36C23, C4×C28 [×4], C4×C28 [×2], C7×C22⋊C4 [×10], C7×C4⋊C4 [×2], C7×C4⋊C4 [×8], C22×C28 [×3], C22×C28 [×2], D4×C14, D4×C14 [×2], Q8×C14, C2×C4×C28, C7×C42⋊C2 [×2], D4×C28, D4×C28 [×2], Q8×C28, C7×C4⋊D4, C7×C22⋊Q8, C7×C22.D4 [×2], C7×C4.4D4, C7×C42.C2, C7×C42⋊2C2 [×2], C7×C23.36C23
Quotients:
C1, C2 [×15], C22 [×35], C7, C23 [×15], C14 [×15], C4○D4 [×6], C24, C2×C14 [×35], C2×C4○D4 [×3], C22×C14 [×15], C23.36C23, C7×C4○D4 [×6], C23×C14, C14×C4○D4 [×3], C7×C23.36C23
Generators and relations
G = < a,b,c,d,e,f,g | a7=b2=c2=d2=e2=1, f2=d, g2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ebe=bd=db, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, eg=ge, fg=gf >
►On 224 points
Generators in S224
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)(22 23 24 25 26 27 28)(29 30 31 32 33 34 35)(36 37 38 39 40 41 42)(43 44 45 46 47 48 49)(50 51 52 53 54 55 56)(57 58 59 60 61 62 63)(64 65 66 67 68 69 70)(71 72 73 74 75 76 77)(78 79 80 81 82 83 84)(85 86 87 88 89 90 91)(92 93 94 95 96 97 98)(99 100 101 102 103 104 105)(106 107 108 109 110 111 112)(113 114 115 116 117 118 119)(120 121 122 123 124 125 126)(127 128 129 130 131 132 133)(134 135 136 137 138 139 140)(141 142 143 144 145 146 147)(148 149 150 151 152 153 154)(155 156 157 158 159 160 161)(162 163 164 165 166 167 168)(169 170 171 172 173 174 175)(176 177 178 179 180 181 182)(183 184 185 186 187 188 189)(190 191 192 193 194 195 196)(197 198 199 200 201 202 203)(204 205 206 207 208 209 210)(211 212 213 214 215 216 217)(218 219 220 221 222 223 224)
(1 34)(2 35)(3 29)(4 30)(5 31)(6 32)(7 33)(8 28)(9 22)(10 23)(11 24)(12 25)(13 26)(14 27)(15 223)(16 224)(17 218)(18 219)(19 220)(20 221)(21 222)(36 43)(37 44)(38 45)(39 46)(40 47)(41 48)(42 49)(50 74)(51 75)(52 76)(53 77)(54 71)(55 72)(56 73)(57 68)(58 69)(59 70)(60 64)(61 65)(62 66)(63 67)(78 102)(79 103)(80 104)(81 105)(82 99)(83 100)(84 101)(85 96)(86 97)(87 98)(88 92)(89 93)(90 94)(91 95)(106 130)(107 131)(108 132)(109 133)(110 127)(111 128)(112 129)(113 124)(114 125)(115 126)(116 120)(117 121)(118 122)(119 123)(134 147)(135 141)(136 142)(137 143)(138 144)(139 145)(140 146)(148 155)(149 156)(150 157)(151 158)(152 159)(153 160)(154 161)(162 175)(163 169)(164 170)(165 171)(166 172)(167 173)(168 174)(176 183)(177 184)(178 185)(179 186)(180 187)(181 188)(182 189)(190 203)(191 197)(192 198)(193 199)(194 200)(195 201)(196 202)(204 211)(205 212)(206 213)(207 214)(208 215)(209 216)(210 217)
(1 34)(2 35)(3 29)(4 30)(5 31)(6 32)(7 33)(8 222)(9 223)(10 224)(11 218)(12 219)(13 220)(14 221)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(36 43)(37 44)(38 45)(39 46)(40 47)(41 48)(42 49)(50 74)(51 75)(52 76)(53 77)(54 71)(55 72)(56 73)(57 68)(58 69)(59 70)(60 64)(61 65)(62 66)(63 67)(78 102)(79 103)(80 104)(81 105)(82 99)(83 100)(84 101)(85 96)(86 97)(87 98)(88 92)(89 93)(90 94)(91 95)(106 130)(107 131)(108 132)(109 133)(110 127)(111 128)(112 129)(113 124)(114 125)(115 126)(116 120)(117 121)(118 122)(119 123)(134 158)(135 159)(136 160)(137 161)(138 155)(139 156)(140 157)(141 152)(142 153)(143 154)(144 148)(145 149)(146 150)(147 151)(162 186)(163 187)(164 188)(165 189)(166 183)(167 184)(168 185)(169 180)(170 181)(171 182)(172 176)(173 177)(174 178)(175 179)(190 214)(191 215)(192 216)(193 217)(194 211)(195 212)(196 213)(197 208)(198 209)(199 210)(200 204)(201 205)(202 206)(203 207)
(1 46)(2 47)(3 48)(4 49)(5 43)(6 44)(7 45)(8 21)(9 15)(10 16)(11 17)(12 18)(13 19)(14 20)(22 223)(23 224)(24 218)(25 219)(26 220)(27 221)(28 222)(29 41)(30 42)(31 36)(32 37)(33 38)(34 39)(35 40)(50 67)(51 68)(52 69)(53 70)(54 64)(55 65)(56 66)(57 75)(58 76)(59 77)(60 71)(61 72)(62 73)(63 74)(78 95)(79 96)(80 97)(81 98)(82 92)(83 93)(84 94)(85 103)(86 104)(87 105)(88 99)(89 100)(90 101)(91 102)(106 123)(107 124)(108 125)(109 126)(110 120)(111 121)(112 122)(113 131)(114 132)(115 133)(116 127)(117 128)(118 129)(119 130)(134 151)(135 152)(136 153)(137 154)(138 148)(139 149)(140 150)(141 159)(142 160)(143 161)(144 155)(145 156)(146 157)(147 158)(162 179)(163 180)(164 181)(165 182)(166 176)(167 177)(168 178)(169 187)(170 188)(171 189)(172 183)(173 184)(174 185)(175 186)(190 207)(191 208)(192 209)(193 210)(194 204)(195 205)(196 206)(197 215)(198 216)(199 217)(200 211)(201 212)(202 213)(203 214)
(1 158)(2 159)(3 160)(4 161)(5 155)(6 156)(7 157)(8 127)(9 128)(10 129)(11 130)(12 131)(13 132)(14 133)(15 117)(16 118)(17 119)(18 113)(19 114)(20 115)(21 116)(22 121)(23 122)(24 123)(25 124)(26 125)(27 126)(28 120)(29 136)(30 137)(31 138)(32 139)(33 140)(34 134)(35 135)(36 148)(37 149)(38 150)(39 151)(40 152)(41 153)(42 154)(43 144)(44 145)(45 146)(46 147)(47 141)(48 142)(49 143)(50 162)(51 163)(52 164)(53 165)(54 166)(55 167)(56 168)(57 169)(58 170)(59 171)(60 172)(61 173)(62 174)(63 175)(64 176)(65 177)(66 178)(67 179)(68 180)(69 181)(70 182)(71 183)(72 184)(73 185)(74 186)(75 187)(76 188)(77 189)(78 190)(79 191)(80 192)(81 193)(82 194)(83 195)(84 196)(85 197)(86 198)(87 199)(88 200)(89 201)(90 202)(91 203)(92 204)(93 205)(94 206)(95 207)(96 208)(97 209)(98 210)(99 211)(100 212)(101 213)(102 214)(103 215)(104 216)(105 217)(106 218)(107 219)(108 220)(109 221)(110 222)(111 223)(112 224)
(1 91 46 102)(2 85 47 103)(3 86 48 104)(4 87 49 105)(5 88 43 99)(6 89 44 100)(7 90 45 101)(8 166 21 176)(9 167 15 177)(10 168 16 178)(11 162 17 179)(12 163 18 180)(13 164 19 181)(14 165 20 182)(22 173 223 184)(23 174 224 185)(24 175 218 186)(25 169 219 187)(26 170 220 188)(27 171 221 189)(28 172 222 183)(29 97 41 80)(30 98 42 81)(31 92 36 82)(32 93 37 83)(33 94 38 84)(34 95 39 78)(35 96 40 79)(50 123 67 106)(51 124 68 107)(52 125 69 108)(53 126 70 109)(54 120 64 110)(55 121 65 111)(56 122 66 112)(57 131 75 113)(58 132 76 114)(59 133 77 115)(60 127 71 116)(61 128 72 117)(62 129 73 118)(63 130 74 119)(134 203 151 214)(135 197 152 215)(136 198 153 216)(137 199 154 217)(138 200 148 211)(139 201 149 212)(140 202 150 213)(141 191 159 208)(142 192 160 209)(143 193 161 210)(144 194 155 204)(145 195 156 205)(146 196 157 206)(147 190 158 207)
(1 50 34 74)(2 51 35 75)(3 52 29 76)(4 53 30 77)(5 54 31 71)(6 55 32 72)(7 56 33 73)(8 211 222 194)(9 212 223 195)(10 213 224 196)(11 214 218 190)(12 215 219 191)(13 216 220 192)(14 217 221 193)(15 201 22 205)(16 202 23 206)(17 203 24 207)(18 197 25 208)(19 198 26 209)(20 199 27 210)(21 200 28 204)(36 60 43 64)(37 61 44 65)(38 62 45 66)(39 63 46 67)(40 57 47 68)(41 58 48 69)(42 59 49 70)(78 130 102 106)(79 131 103 107)(80 132 104 108)(81 133 105 109)(82 127 99 110)(83 128 100 111)(84 129 101 112)(85 124 96 113)(86 125 97 114)(87 126 98 115)(88 120 92 116)(89 121 93 117)(90 122 94 118)(91 123 95 119)(134 186 158 162)(135 187 159 163)(136 188 160 164)(137 189 161 165)(138 183 155 166)(139 184 156 167)(140 185 157 168)(141 180 152 169)(142 181 153 170)(143 182 154 171)(144 176 148 172)(145 177 149 173)(146 178 150 174)(147 179 151 175)
G:=sub<Sym(224)| (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28)(29,30,31,32,33,34,35)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49)(50,51,52,53,54,55,56)(57,58,59,60,61,62,63)(64,65,66,67,68,69,70)(71,72,73,74,75,76,77)(78,79,80,81,82,83,84)(85,86,87,88,89,90,91)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112)(113,114,115,116,117,118,119)(120,121,122,123,124,125,126)(127,128,129,130,131,132,133)(134,135,136,137,138,139,140)(141,142,143,144,145,146,147)(148,149,150,151,152,153,154)(155,156,157,158,159,160,161)(162,163,164,165,166,167,168)(169,170,171,172,173,174,175)(176,177,178,179,180,181,182)(183,184,185,186,187,188,189)(190,191,192,193,194,195,196)(197,198,199,200,201,202,203)(204,205,206,207,208,209,210)(211,212,213,214,215,216,217)(218,219,220,221,222,223,224), (1,34)(2,35)(3,29)(4,30)(5,31)(6,32)(7,33)(8,28)(9,22)(10,23)(11,24)(12,25)(13,26)(14,27)(15,223)(16,224)(17,218)(18,219)(19,220)(20,221)(21,222)(36,43)(37,44)(38,45)(39,46)(40,47)(41,48)(42,49)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,68)(58,69)(59,70)(60,64)(61,65)(62,66)(63,67)(78,102)(79,103)(80,104)(81,105)(82,99)(83,100)(84,101)(85,96)(86,97)(87,98)(88,92)(89,93)(90,94)(91,95)(106,130)(107,131)(108,132)(109,133)(110,127)(111,128)(112,129)(113,124)(114,125)(115,126)(116,120)(117,121)(118,122)(119,123)(134,147)(135,141)(136,142)(137,143)(138,144)(139,145)(140,146)(148,155)(149,156)(150,157)(151,158)(152,159)(153,160)(154,161)(162,175)(163,169)(164,170)(165,171)(166,172)(167,173)(168,174)(176,183)(177,184)(178,185)(179,186)(180,187)(181,188)(182,189)(190,203)(191,197)(192,198)(193,199)(194,200)(195,201)(196,202)(204,211)(205,212)(206,213)(207,214)(208,215)(209,216)(210,217), (1,34)(2,35)(3,29)(4,30)(5,31)(6,32)(7,33)(8,222)(9,223)(10,224)(11,218)(12,219)(13,220)(14,221)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(36,43)(37,44)(38,45)(39,46)(40,47)(41,48)(42,49)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,68)(58,69)(59,70)(60,64)(61,65)(62,66)(63,67)(78,102)(79,103)(80,104)(81,105)(82,99)(83,100)(84,101)(85,96)(86,97)(87,98)(88,92)(89,93)(90,94)(91,95)(106,130)(107,131)(108,132)(109,133)(110,127)(111,128)(112,129)(113,124)(114,125)(115,126)(116,120)(117,121)(118,122)(119,123)(134,158)(135,159)(136,160)(137,161)(138,155)(139,156)(140,157)(141,152)(142,153)(143,154)(144,148)(145,149)(146,150)(147,151)(162,186)(163,187)(164,188)(165,189)(166,183)(167,184)(168,185)(169,180)(170,181)(171,182)(172,176)(173,177)(174,178)(175,179)(190,214)(191,215)(192,216)(193,217)(194,211)(195,212)(196,213)(197,208)(198,209)(199,210)(200,204)(201,205)(202,206)(203,207), (1,46)(2,47)(3,48)(4,49)(5,43)(6,44)(7,45)(8,21)(9,15)(10,16)(11,17)(12,18)(13,19)(14,20)(22,223)(23,224)(24,218)(25,219)(26,220)(27,221)(28,222)(29,41)(30,42)(31,36)(32,37)(33,38)(34,39)(35,40)(50,67)(51,68)(52,69)(53,70)(54,64)(55,65)(56,66)(57,75)(58,76)(59,77)(60,71)(61,72)(62,73)(63,74)(78,95)(79,96)(80,97)(81,98)(82,92)(83,93)(84,94)(85,103)(86,104)(87,105)(88,99)(89,100)(90,101)(91,102)(106,123)(107,124)(108,125)(109,126)(110,120)(111,121)(112,122)(113,131)(114,132)(115,133)(116,127)(117,128)(118,129)(119,130)(134,151)(135,152)(136,153)(137,154)(138,148)(139,149)(140,150)(141,159)(142,160)(143,161)(144,155)(145,156)(146,157)(147,158)(162,179)(163,180)(164,181)(165,182)(166,176)(167,177)(168,178)(169,187)(170,188)(171,189)(172,183)(173,184)(174,185)(175,186)(190,207)(191,208)(192,209)(193,210)(194,204)(195,205)(196,206)(197,215)(198,216)(199,217)(200,211)(201,212)(202,213)(203,214), (1,158)(2,159)(3,160)(4,161)(5,155)(6,156)(7,157)(8,127)(9,128)(10,129)(11,130)(12,131)(13,132)(14,133)(15,117)(16,118)(17,119)(18,113)(19,114)(20,115)(21,116)(22,121)(23,122)(24,123)(25,124)(26,125)(27,126)(28,120)(29,136)(30,137)(31,138)(32,139)(33,140)(34,134)(35,135)(36,148)(37,149)(38,150)(39,151)(40,152)(41,153)(42,154)(43,144)(44,145)(45,146)(46,147)(47,141)(48,142)(49,143)(50,162)(51,163)(52,164)(53,165)(54,166)(55,167)(56,168)(57,169)(58,170)(59,171)(60,172)(61,173)(62,174)(63,175)(64,176)(65,177)(66,178)(67,179)(68,180)(69,181)(70,182)(71,183)(72,184)(73,185)(74,186)(75,187)(76,188)(77,189)(78,190)(79,191)(80,192)(81,193)(82,194)(83,195)(84,196)(85,197)(86,198)(87,199)(88,200)(89,201)(90,202)(91,203)(92,204)(93,205)(94,206)(95,207)(96,208)(97,209)(98,210)(99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(106,218)(107,219)(108,220)(109,221)(110,222)(111,223)(112,224), (1,91,46,102)(2,85,47,103)(3,86,48,104)(4,87,49,105)(5,88,43,99)(6,89,44,100)(7,90,45,101)(8,166,21,176)(9,167,15,177)(10,168,16,178)(11,162,17,179)(12,163,18,180)(13,164,19,181)(14,165,20,182)(22,173,223,184)(23,174,224,185)(24,175,218,186)(25,169,219,187)(26,170,220,188)(27,171,221,189)(28,172,222,183)(29,97,41,80)(30,98,42,81)(31,92,36,82)(32,93,37,83)(33,94,38,84)(34,95,39,78)(35,96,40,79)(50,123,67,106)(51,124,68,107)(52,125,69,108)(53,126,70,109)(54,120,64,110)(55,121,65,111)(56,122,66,112)(57,131,75,113)(58,132,76,114)(59,133,77,115)(60,127,71,116)(61,128,72,117)(62,129,73,118)(63,130,74,119)(134,203,151,214)(135,197,152,215)(136,198,153,216)(137,199,154,217)(138,200,148,211)(139,201,149,212)(140,202,150,213)(141,191,159,208)(142,192,160,209)(143,193,161,210)(144,194,155,204)(145,195,156,205)(146,196,157,206)(147,190,158,207), (1,50,34,74)(2,51,35,75)(3,52,29,76)(4,53,30,77)(5,54,31,71)(6,55,32,72)(7,56,33,73)(8,211,222,194)(9,212,223,195)(10,213,224,196)(11,214,218,190)(12,215,219,191)(13,216,220,192)(14,217,221,193)(15,201,22,205)(16,202,23,206)(17,203,24,207)(18,197,25,208)(19,198,26,209)(20,199,27,210)(21,200,28,204)(36,60,43,64)(37,61,44,65)(38,62,45,66)(39,63,46,67)(40,57,47,68)(41,58,48,69)(42,59,49,70)(78,130,102,106)(79,131,103,107)(80,132,104,108)(81,133,105,109)(82,127,99,110)(83,128,100,111)(84,129,101,112)(85,124,96,113)(86,125,97,114)(87,126,98,115)(88,120,92,116)(89,121,93,117)(90,122,94,118)(91,123,95,119)(134,186,158,162)(135,187,159,163)(136,188,160,164)(137,189,161,165)(138,183,155,166)(139,184,156,167)(140,185,157,168)(141,180,152,169)(142,181,153,170)(143,182,154,171)(144,176,148,172)(145,177,149,173)(146,178,150,174)(147,179,151,175)>;
G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28)(29,30,31,32,33,34,35)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49)(50,51,52,53,54,55,56)(57,58,59,60,61,62,63)(64,65,66,67,68,69,70)(71,72,73,74,75,76,77)(78,79,80,81,82,83,84)(85,86,87,88,89,90,91)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112)(113,114,115,116,117,118,119)(120,121,122,123,124,125,126)(127,128,129,130,131,132,133)(134,135,136,137,138,139,140)(141,142,143,144,145,146,147)(148,149,150,151,152,153,154)(155,156,157,158,159,160,161)(162,163,164,165,166,167,168)(169,170,171,172,173,174,175)(176,177,178,179,180,181,182)(183,184,185,186,187,188,189)(190,191,192,193,194,195,196)(197,198,199,200,201,202,203)(204,205,206,207,208,209,210)(211,212,213,214,215,216,217)(218,219,220,221,222,223,224), (1,34)(2,35)(3,29)(4,30)(5,31)(6,32)(7,33)(8,28)(9,22)(10,23)(11,24)(12,25)(13,26)(14,27)(15,223)(16,224)(17,218)(18,219)(19,220)(20,221)(21,222)(36,43)(37,44)(38,45)(39,46)(40,47)(41,48)(42,49)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,68)(58,69)(59,70)(60,64)(61,65)(62,66)(63,67)(78,102)(79,103)(80,104)(81,105)(82,99)(83,100)(84,101)(85,96)(86,97)(87,98)(88,92)(89,93)(90,94)(91,95)(106,130)(107,131)(108,132)(109,133)(110,127)(111,128)(112,129)(113,124)(114,125)(115,126)(116,120)(117,121)(118,122)(119,123)(134,147)(135,141)(136,142)(137,143)(138,144)(139,145)(140,146)(148,155)(149,156)(150,157)(151,158)(152,159)(153,160)(154,161)(162,175)(163,169)(164,170)(165,171)(166,172)(167,173)(168,174)(176,183)(177,184)(178,185)(179,186)(180,187)(181,188)(182,189)(190,203)(191,197)(192,198)(193,199)(194,200)(195,201)(196,202)(204,211)(205,212)(206,213)(207,214)(208,215)(209,216)(210,217), (1,34)(2,35)(3,29)(4,30)(5,31)(6,32)(7,33)(8,222)(9,223)(10,224)(11,218)(12,219)(13,220)(14,221)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(36,43)(37,44)(38,45)(39,46)(40,47)(41,48)(42,49)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,68)(58,69)(59,70)(60,64)(61,65)(62,66)(63,67)(78,102)(79,103)(80,104)(81,105)(82,99)(83,100)(84,101)(85,96)(86,97)(87,98)(88,92)(89,93)(90,94)(91,95)(106,130)(107,131)(108,132)(109,133)(110,127)(111,128)(112,129)(113,124)(114,125)(115,126)(116,120)(117,121)(118,122)(119,123)(134,158)(135,159)(136,160)(137,161)(138,155)(139,156)(140,157)(141,152)(142,153)(143,154)(144,148)(145,149)(146,150)(147,151)(162,186)(163,187)(164,188)(165,189)(166,183)(167,184)(168,185)(169,180)(170,181)(171,182)(172,176)(173,177)(174,178)(175,179)(190,214)(191,215)(192,216)(193,217)(194,211)(195,212)(196,213)(197,208)(198,209)(199,210)(200,204)(201,205)(202,206)(203,207), (1,46)(2,47)(3,48)(4,49)(5,43)(6,44)(7,45)(8,21)(9,15)(10,16)(11,17)(12,18)(13,19)(14,20)(22,223)(23,224)(24,218)(25,219)(26,220)(27,221)(28,222)(29,41)(30,42)(31,36)(32,37)(33,38)(34,39)(35,40)(50,67)(51,68)(52,69)(53,70)(54,64)(55,65)(56,66)(57,75)(58,76)(59,77)(60,71)(61,72)(62,73)(63,74)(78,95)(79,96)(80,97)(81,98)(82,92)(83,93)(84,94)(85,103)(86,104)(87,105)(88,99)(89,100)(90,101)(91,102)(106,123)(107,124)(108,125)(109,126)(110,120)(111,121)(112,122)(113,131)(114,132)(115,133)(116,127)(117,128)(118,129)(119,130)(134,151)(135,152)(136,153)(137,154)(138,148)(139,149)(140,150)(141,159)(142,160)(143,161)(144,155)(145,156)(146,157)(147,158)(162,179)(163,180)(164,181)(165,182)(166,176)(167,177)(168,178)(169,187)(170,188)(171,189)(172,183)(173,184)(174,185)(175,186)(190,207)(191,208)(192,209)(193,210)(194,204)(195,205)(196,206)(197,215)(198,216)(199,217)(200,211)(201,212)(202,213)(203,214), (1,158)(2,159)(3,160)(4,161)(5,155)(6,156)(7,157)(8,127)(9,128)(10,129)(11,130)(12,131)(13,132)(14,133)(15,117)(16,118)(17,119)(18,113)(19,114)(20,115)(21,116)(22,121)(23,122)(24,123)(25,124)(26,125)(27,126)(28,120)(29,136)(30,137)(31,138)(32,139)(33,140)(34,134)(35,135)(36,148)(37,149)(38,150)(39,151)(40,152)(41,153)(42,154)(43,144)(44,145)(45,146)(46,147)(47,141)(48,142)(49,143)(50,162)(51,163)(52,164)(53,165)(54,166)(55,167)(56,168)(57,169)(58,170)(59,171)(60,172)(61,173)(62,174)(63,175)(64,176)(65,177)(66,178)(67,179)(68,180)(69,181)(70,182)(71,183)(72,184)(73,185)(74,186)(75,187)(76,188)(77,189)(78,190)(79,191)(80,192)(81,193)(82,194)(83,195)(84,196)(85,197)(86,198)(87,199)(88,200)(89,201)(90,202)(91,203)(92,204)(93,205)(94,206)(95,207)(96,208)(97,209)(98,210)(99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(106,218)(107,219)(108,220)(109,221)(110,222)(111,223)(112,224), (1,91,46,102)(2,85,47,103)(3,86,48,104)(4,87,49,105)(5,88,43,99)(6,89,44,100)(7,90,45,101)(8,166,21,176)(9,167,15,177)(10,168,16,178)(11,162,17,179)(12,163,18,180)(13,164,19,181)(14,165,20,182)(22,173,223,184)(23,174,224,185)(24,175,218,186)(25,169,219,187)(26,170,220,188)(27,171,221,189)(28,172,222,183)(29,97,41,80)(30,98,42,81)(31,92,36,82)(32,93,37,83)(33,94,38,84)(34,95,39,78)(35,96,40,79)(50,123,67,106)(51,124,68,107)(52,125,69,108)(53,126,70,109)(54,120,64,110)(55,121,65,111)(56,122,66,112)(57,131,75,113)(58,132,76,114)(59,133,77,115)(60,127,71,116)(61,128,72,117)(62,129,73,118)(63,130,74,119)(134,203,151,214)(135,197,152,215)(136,198,153,216)(137,199,154,217)(138,200,148,211)(139,201,149,212)(140,202,150,213)(141,191,159,208)(142,192,160,209)(143,193,161,210)(144,194,155,204)(145,195,156,205)(146,196,157,206)(147,190,158,207), (1,50,34,74)(2,51,35,75)(3,52,29,76)(4,53,30,77)(5,54,31,71)(6,55,32,72)(7,56,33,73)(8,211,222,194)(9,212,223,195)(10,213,224,196)(11,214,218,190)(12,215,219,191)(13,216,220,192)(14,217,221,193)(15,201,22,205)(16,202,23,206)(17,203,24,207)(18,197,25,208)(19,198,26,209)(20,199,27,210)(21,200,28,204)(36,60,43,64)(37,61,44,65)(38,62,45,66)(39,63,46,67)(40,57,47,68)(41,58,48,69)(42,59,49,70)(78,130,102,106)(79,131,103,107)(80,132,104,108)(81,133,105,109)(82,127,99,110)(83,128,100,111)(84,129,101,112)(85,124,96,113)(86,125,97,114)(87,126,98,115)(88,120,92,116)(89,121,93,117)(90,122,94,118)(91,123,95,119)(134,186,158,162)(135,187,159,163)(136,188,160,164)(137,189,161,165)(138,183,155,166)(139,184,156,167)(140,185,157,168)(141,180,152,169)(142,181,153,170)(143,182,154,171)(144,176,148,172)(145,177,149,173)(146,178,150,174)(147,179,151,175) );
G=PermutationGroup([(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21),(22,23,24,25,26,27,28),(29,30,31,32,33,34,35),(36,37,38,39,40,41,42),(43,44,45,46,47,48,49),(50,51,52,53,54,55,56),(57,58,59,60,61,62,63),(64,65,66,67,68,69,70),(71,72,73,74,75,76,77),(78,79,80,81,82,83,84),(85,86,87,88,89,90,91),(92,93,94,95,96,97,98),(99,100,101,102,103,104,105),(106,107,108,109,110,111,112),(113,114,115,116,117,118,119),(120,121,122,123,124,125,126),(127,128,129,130,131,132,133),(134,135,136,137,138,139,140),(141,142,143,144,145,146,147),(148,149,150,151,152,153,154),(155,156,157,158,159,160,161),(162,163,164,165,166,167,168),(169,170,171,172,173,174,175),(176,177,178,179,180,181,182),(183,184,185,186,187,188,189),(190,191,192,193,194,195,196),(197,198,199,200,201,202,203),(204,205,206,207,208,209,210),(211,212,213,214,215,216,217),(218,219,220,221,222,223,224)], [(1,34),(2,35),(3,29),(4,30),(5,31),(6,32),(7,33),(8,28),(9,22),(10,23),(11,24),(12,25),(13,26),(14,27),(15,223),(16,224),(17,218),(18,219),(19,220),(20,221),(21,222),(36,43),(37,44),(38,45),(39,46),(40,47),(41,48),(42,49),(50,74),(51,75),(52,76),(53,77),(54,71),(55,72),(56,73),(57,68),(58,69),(59,70),(60,64),(61,65),(62,66),(63,67),(78,102),(79,103),(80,104),(81,105),(82,99),(83,100),(84,101),(85,96),(86,97),(87,98),(88,92),(89,93),(90,94),(91,95),(106,130),(107,131),(108,132),(109,133),(110,127),(111,128),(112,129),(113,124),(114,125),(115,126),(116,120),(117,121),(118,122),(119,123),(134,147),(135,141),(136,142),(137,143),(138,144),(139,145),(140,146),(148,155),(149,156),(150,157),(151,158),(152,159),(153,160),(154,161),(162,175),(163,169),(164,170),(165,171),(166,172),(167,173),(168,174),(176,183),(177,184),(178,185),(179,186),(180,187),(181,188),(182,189),(190,203),(191,197),(192,198),(193,199),(194,200),(195,201),(196,202),(204,211),(205,212),(206,213),(207,214),(208,215),(209,216),(210,217)], [(1,34),(2,35),(3,29),(4,30),(5,31),(6,32),(7,33),(8,222),(9,223),(10,224),(11,218),(12,219),(13,220),(14,221),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(36,43),(37,44),(38,45),(39,46),(40,47),(41,48),(42,49),(50,74),(51,75),(52,76),(53,77),(54,71),(55,72),(56,73),(57,68),(58,69),(59,70),(60,64),(61,65),(62,66),(63,67),(78,102),(79,103),(80,104),(81,105),(82,99),(83,100),(84,101),(85,96),(86,97),(87,98),(88,92),(89,93),(90,94),(91,95),(106,130),(107,131),(108,132),(109,133),(110,127),(111,128),(112,129),(113,124),(114,125),(115,126),(116,120),(117,121),(118,122),(119,123),(134,158),(135,159),(136,160),(137,161),(138,155),(139,156),(140,157),(141,152),(142,153),(143,154),(144,148),(145,149),(146,150),(147,151),(162,186),(163,187),(164,188),(165,189),(166,183),(167,184),(168,185),(169,180),(170,181),(171,182),(172,176),(173,177),(174,178),(175,179),(190,214),(191,215),(192,216),(193,217),(194,211),(195,212),(196,213),(197,208),(198,209),(199,210),(200,204),(201,205),(202,206),(203,207)], [(1,46),(2,47),(3,48),(4,49),(5,43),(6,44),(7,45),(8,21),(9,15),(10,16),(11,17),(12,18),(13,19),(14,20),(22,223),(23,224),(24,218),(25,219),(26,220),(27,221),(28,222),(29,41),(30,42),(31,36),(32,37),(33,38),(34,39),(35,40),(50,67),(51,68),(52,69),(53,70),(54,64),(55,65),(56,66),(57,75),(58,76),(59,77),(60,71),(61,72),(62,73),(63,74),(78,95),(79,96),(80,97),(81,98),(82,92),(83,93),(84,94),(85,103),(86,104),(87,105),(88,99),(89,100),(90,101),(91,102),(106,123),(107,124),(108,125),(109,126),(110,120),(111,121),(112,122),(113,131),(114,132),(115,133),(116,127),(117,128),(118,129),(119,130),(134,151),(135,152),(136,153),(137,154),(138,148),(139,149),(140,150),(141,159),(142,160),(143,161),(144,155),(145,156),(146,157),(147,158),(162,179),(163,180),(164,181),(165,182),(166,176),(167,177),(168,178),(169,187),(170,188),(171,189),(172,183),(173,184),(174,185),(175,186),(190,207),(191,208),(192,209),(193,210),(194,204),(195,205),(196,206),(197,215),(198,216),(199,217),(200,211),(201,212),(202,213),(203,214)], [(1,158),(2,159),(3,160),(4,161),(5,155),(6,156),(7,157),(8,127),(9,128),(10,129),(11,130),(12,131),(13,132),(14,133),(15,117),(16,118),(17,119),(18,113),(19,114),(20,115),(21,116),(22,121),(23,122),(24,123),(25,124),(26,125),(27,126),(28,120),(29,136),(30,137),(31,138),(32,139),(33,140),(34,134),(35,135),(36,148),(37,149),(38,150),(39,151),(40,152),(41,153),(42,154),(43,144),(44,145),(45,146),(46,147),(47,141),(48,142),(49,143),(50,162),(51,163),(52,164),(53,165),(54,166),(55,167),(56,168),(57,169),(58,170),(59,171),(60,172),(61,173),(62,174),(63,175),(64,176),(65,177),(66,178),(67,179),(68,180),(69,181),(70,182),(71,183),(72,184),(73,185),(74,186),(75,187),(76,188),(77,189),(78,190),(79,191),(80,192),(81,193),(82,194),(83,195),(84,196),(85,197),(86,198),(87,199),(88,200),(89,201),(90,202),(91,203),(92,204),(93,205),(94,206),(95,207),(96,208),(97,209),(98,210),(99,211),(100,212),(101,213),(102,214),(103,215),(104,216),(105,217),(106,218),(107,219),(108,220),(109,221),(110,222),(111,223),(112,224)], [(1,91,46,102),(2,85,47,103),(3,86,48,104),(4,87,49,105),(5,88,43,99),(6,89,44,100),(7,90,45,101),(8,166,21,176),(9,167,15,177),(10,168,16,178),(11,162,17,179),(12,163,18,180),(13,164,19,181),(14,165,20,182),(22,173,223,184),(23,174,224,185),(24,175,218,186),(25,169,219,187),(26,170,220,188),(27,171,221,189),(28,172,222,183),(29,97,41,80),(30,98,42,81),(31,92,36,82),(32,93,37,83),(33,94,38,84),(34,95,39,78),(35,96,40,79),(50,123,67,106),(51,124,68,107),(52,125,69,108),(53,126,70,109),(54,120,64,110),(55,121,65,111),(56,122,66,112),(57,131,75,113),(58,132,76,114),(59,133,77,115),(60,127,71,116),(61,128,72,117),(62,129,73,118),(63,130,74,119),(134,203,151,214),(135,197,152,215),(136,198,153,216),(137,199,154,217),(138,200,148,211),(139,201,149,212),(140,202,150,213),(141,191,159,208),(142,192,160,209),(143,193,161,210),(144,194,155,204),(145,195,156,205),(146,196,157,206),(147,190,158,207)], [(1,50,34,74),(2,51,35,75),(3,52,29,76),(4,53,30,77),(5,54,31,71),(6,55,32,72),(7,56,33,73),(8,211,222,194),(9,212,223,195),(10,213,224,196),(11,214,218,190),(12,215,219,191),(13,216,220,192),(14,217,221,193),(15,201,22,205),(16,202,23,206),(17,203,24,207),(18,197,25,208),(19,198,26,209),(20,199,27,210),(21,200,28,204),(36,60,43,64),(37,61,44,65),(38,62,45,66),(39,63,46,67),(40,57,47,68),(41,58,48,69),(42,59,49,70),(78,130,102,106),(79,131,103,107),(80,132,104,108),(81,133,105,109),(82,127,99,110),(83,128,100,111),(84,129,101,112),(85,124,96,113),(86,125,97,114),(87,126,98,115),(88,120,92,116),(89,121,93,117),(90,122,94,118),(91,123,95,119),(134,186,158,162),(135,187,159,163),(136,188,160,164),(137,189,161,165),(138,183,155,166),(139,184,156,167),(140,185,157,168),(141,180,152,169),(142,181,153,170),(143,182,154,171),(144,176,148,172),(145,177,149,173),(146,178,150,174),(147,179,151,175)])
Matrix representation ►G ⊆ GL4(𝔽29) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 24 | 0 |
| 0 | 0 | 0 | 24 |
| 28 | 8 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 28 | 28 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 21 | 17 | 0 | 0 |
| 27 | 8 | 0 | 0 |
| 0 | 0 | 28 | 27 |
| 0 | 0 | 0 | 1 |
| 17 | 9 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
G:=sub<GL(4,GF(29))| [16,0,0,0,0,16,0,0,0,0,24,0,0,0,0,24],[28,0,0,0,8,1,0,0,0,0,1,28,0,0,0,28],[28,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[21,27,0,0,17,8,0,0,0,0,28,0,0,0,27,1],[17,0,0,0,9,12,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,28,0,0,0,0,28] >;
196 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4T | 7A | ··· | 7F | 14A | ··· | 14R | 14S | ··· | 14AD | 14AE | ··· | 14AP | 28A | ··· | 28X | 28Y | ··· | 28CF | 28CG | ··· | 28DP |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
196 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | C14 | C14 | C14 | C14 | C14 | C4○D4 | C4○D4 | C7×C4○D4 | C7×C4○D4 |
| kernel | C7×C23.36C23 | C2×C4×C28 | C7×C42⋊C2 | D4×C28 | Q8×C28 | C7×C4⋊D4 | C7×C22⋊Q8 | C7×C22.D4 | C7×C4.4D4 | C7×C42.C2 | C7×C42⋊2C2 | C23.36C23 | C2×C42 | C42⋊C2 | C4×D4 | C4×Q8 | C4⋊D4 | C22⋊Q8 | C22.D4 | C4.4D4 | C42.C2 | C42⋊2C2 | C28 | C2×C14 | C4 | C22 |
| # reps | 1 | 1 | 2 | 3 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 6 | 6 | 12 | 18 | 6 | 6 | 6 | 12 | 6 | 6 | 12 | 8 | 4 | 48 | 24 |
In GAP, Magma, Sage, TeX
C_7\times C_2^3._{36}C_2^3 % in TeX
G:=Group("C7xC2^3.36C2^3"); // GroupNames label
G:=SmallGroup(448,1312);
// by ID
G=gap.SmallGroup(448,1312);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1597,1576,4790,416]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^7=b^2=c^2=d^2=e^2=1,f^2=d,g^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,e*b*e=b*d=d*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f^-1=c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,e*g=g*e,f*g=g*f>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀