direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C22.26C24, (C4×D4)⋊9C14, (C2×C28)⋊34D4, C4⋊Q8⋊19C14, (D4×C28)⋊38C2, C4.15(D4×C14), C28⋊13(C4○D4), C4⋊D4⋊20C14, C4⋊1D4⋊10C14, (C2×C42)⋊12C14, C28.322(C2×D4), C22.1(D4×C14), C4.4D4⋊18C14, C42.90(C2×C14), (C2×C28).661C23, (C2×C14).352C24, (C4×C28).375C22, C14.188(C22×D4), (D4×C14).319C22, (C22×C14).89C23, C22.26(C23×C14), C23.34(C22×C14), (Q8×C14).268C22, (C22×C28).597C22, (C2×C4×C28)⋊25C2, C4⋊1(C7×C4○D4), (C2×C4)⋊8(C7×D4), (C7×C4⋊Q8)⋊40C2, C2.12(D4×C2×C14), (C2×C4○D4)⋊3C14, (C14×C4○D4)⋊19C2, (C7×C4⋊D4)⋊47C2, (C7×C4⋊1D4)⋊21C2, C4⋊C4.66(C2×C14), C2.13(C14×C4○D4), (C2×C14).89(C2×D4), (C2×D4).64(C2×C14), C14.232(C2×C4○D4), (C7×C4.4D4)⋊38C2, (C2×Q8).55(C2×C14), (C7×C4⋊C4).389C22, C22⋊C4.13(C2×C14), (C2×C4).19(C22×C14), (C22×C4).56(C2×C14), (C7×C22⋊C4).147C22, SmallGroup(448,1315)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C22.26C24
G = < a,b,c,d,e,f,g | a7=b2=c2=d2=e2=1, f2=c, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede=bd=db, be=eb, bf=fb, bg=gb, fdf-1=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, eg=ge, fg=gf >
Subgroups: 466 in 310 conjugacy classes, 170 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C14, C14, C14, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C28, C28, C2×C14, C2×C14, C2×C14, C2×C42, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×C4○D4, C2×C28, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C22×C14, C22.26C24, C4×C28, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C22×C28, C22×C28, D4×C14, Q8×C14, C7×C4○D4, C2×C4×C28, D4×C28, C7×C4⋊D4, C7×C4.4D4, C7×C4⋊1D4, C7×C4⋊Q8, C14×C4○D4, C7×C22.26C24
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C4○D4, C24, C2×C14, C22×D4, C2×C4○D4, C7×D4, C22×C14, C22.26C24, D4×C14, C7×C4○D4, C23×C14, D4×C2×C14, C14×C4○D4, C7×C22.26C24
►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 39)(2 40)(3 41)(4 42)(5 36)(6 37)(7 38)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 22)(15 218)(16 219)(17 220)(18 221)(19 222)(20 223)(21 224)(29 44)(30 45)(31 46)(32 47)(33 48)(34 49)(35 43)(50 66)(51 67)(52 68)(53 69)(54 70)(55 64)(56 65)(57 72)(58 73)(59 74)(60 75)(61 76)(62 77)(63 71)(78 94)(79 95)(80 96)(81 97)(82 98)(83 92)(84 93)(85 100)(86 101)(87 102)(88 103)(89 104)(90 105)(91 99)(106 122)(107 123)(108 124)(109 125)(110 126)(111 120)(112 121)(113 128)(114 129)(115 130)(116 131)(117 132)(118 133)(119 127)(134 150)(135 151)(136 152)(137 153)(138 154)(139 148)(140 149)(141 156)(142 157)(143 158)(144 159)(145 160)(146 161)(147 155)(162 178)(163 179)(164 180)(165 181)(166 182)(167 176)(168 177)(169 184)(170 185)(171 186)(172 187)(173 188)(174 189)(175 183)(190 206)(191 207)(192 208)(193 209)(194 210)(195 204)(196 205)(197 212)(198 213)(199 214)(200 215)(201 216)(202 217)(203 211)
(1 35)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(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)(36 47)(37 48)(38 49)(39 43)(40 44)(41 45)(42 46)(50 71)(51 72)(52 73)(53 74)(54 75)(55 76)(56 77)(57 67)(58 68)(59 69)(60 70)(61 64)(62 65)(63 66)(78 99)(79 100)(80 101)(81 102)(82 103)(83 104)(84 105)(85 95)(86 96)(87 97)(88 98)(89 92)(90 93)(91 94)(106 127)(107 128)(108 129)(109 130)(110 131)(111 132)(112 133)(113 123)(114 124)(115 125)(116 126)(117 120)(118 121)(119 122)(134 155)(135 156)(136 157)(137 158)(138 159)(139 160)(140 161)(141 151)(142 152)(143 153)(144 154)(145 148)(146 149)(147 150)(162 183)(163 184)(164 185)(165 186)(166 187)(167 188)(168 189)(169 179)(170 180)(171 181)(172 182)(173 176)(174 177)(175 178)(190 211)(191 212)(192 213)(193 214)(194 215)(195 216)(196 217)(197 207)(198 208)(199 209)(200 210)(201 204)(202 205)(203 206)
(1 150)(2 151)(3 152)(4 153)(5 154)(6 148)(7 149)(8 118)(9 119)(10 113)(11 114)(12 115)(13 116)(14 117)(15 122)(16 123)(17 124)(18 125)(19 126)(20 120)(21 121)(22 132)(23 133)(24 127)(25 128)(26 129)(27 130)(28 131)(29 141)(30 142)(31 143)(32 144)(33 145)(34 146)(35 147)(36 138)(37 139)(38 140)(39 134)(40 135)(41 136)(42 137)(43 155)(44 156)(45 157)(46 158)(47 159)(48 160)(49 161)(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 35)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 224)(9 218)(10 219)(11 220)(12 221)(13 222)(14 223)(15 24)(16 25)(17 26)(18 27)(19 28)(20 22)(21 23)(36 47)(37 48)(38 49)(39 43)(40 44)(41 45)(42 46)(50 71)(51 72)(52 73)(53 74)(54 75)(55 76)(56 77)(57 67)(58 68)(59 69)(60 70)(61 64)(62 65)(63 66)(78 99)(79 100)(80 101)(81 102)(82 103)(83 104)(84 105)(85 95)(86 96)(87 97)(88 98)(89 92)(90 93)(91 94)(106 127)(107 128)(108 129)(109 130)(110 131)(111 132)(112 133)(113 123)(114 124)(115 125)(116 126)(117 120)(118 121)(119 122)(134 147)(135 141)(136 142)(137 143)(138 144)(139 145)(140 146)(148 160)(149 161)(150 155)(151 156)(152 157)(153 158)(154 159)(162 175)(163 169)(164 170)(165 171)(166 172)(167 173)(168 174)(176 188)(177 189)(178 183)(179 184)(180 185)(181 186)(182 187)(190 203)(191 197)(192 198)(193 199)(194 200)(195 201)(196 202)(204 216)(205 217)(206 211)(207 212)(208 213)(209 214)(210 215)
(1 91 35 94)(2 85 29 95)(3 86 30 96)(4 87 31 97)(5 88 32 98)(6 89 33 92)(7 90 34 93)(8 177 21 174)(9 178 15 175)(10 179 16 169)(11 180 17 170)(12 181 18 171)(13 182 19 172)(14 176 20 173)(22 167 223 188)(23 168 224 189)(24 162 218 183)(25 163 219 184)(26 164 220 185)(27 165 221 186)(28 166 222 187)(36 103 47 82)(37 104 48 83)(38 105 49 84)(39 99 43 78)(40 100 44 79)(41 101 45 80)(42 102 46 81)(50 127 71 106)(51 128 72 107)(52 129 73 108)(53 130 74 109)(54 131 75 110)(55 132 76 111)(56 133 77 112)(57 123 67 113)(58 124 68 114)(59 125 69 115)(60 126 70 116)(61 120 64 117)(62 121 65 118)(63 122 66 119)(134 190 155 211)(135 191 156 212)(136 192 157 213)(137 193 158 214)(138 194 159 215)(139 195 160 216)(140 196 161 217)(141 197 151 207)(142 198 152 208)(143 199 153 209)(144 200 154 210)(145 201 148 204)(146 202 149 205)(147 203 150 206)
(1 50 39 66)(2 51 40 67)(3 52 41 68)(4 53 42 69)(5 54 36 70)(6 55 37 64)(7 56 38 65)(8 202 23 217)(9 203 24 211)(10 197 25 212)(11 198 26 213)(12 199 27 214)(13 200 28 215)(14 201 22 216)(15 206 218 190)(16 207 219 191)(17 208 220 192)(18 209 221 193)(19 210 222 194)(20 204 223 195)(21 205 224 196)(29 72 44 57)(30 73 45 58)(31 74 46 59)(32 75 47 60)(33 76 48 61)(34 77 49 62)(35 71 43 63)(78 122 94 106)(79 123 95 107)(80 124 96 108)(81 125 97 109)(82 126 98 110)(83 120 92 111)(84 121 93 112)(85 128 100 113)(86 129 101 114)(87 130 102 115)(88 131 103 116)(89 132 104 117)(90 133 105 118)(91 127 99 119)(134 178 150 162)(135 179 151 163)(136 180 152 164)(137 181 153 165)(138 182 154 166)(139 176 148 167)(140 177 149 168)(141 184 156 169)(142 185 157 170)(143 186 158 171)(144 187 159 172)(145 188 160 173)(146 189 161 174)(147 183 155 175)
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), (1,39)(2,40)(3,41)(4,42)(5,36)(6,37)(7,38)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,22)(15,218)(16,219)(17,220)(18,221)(19,222)(20,223)(21,224)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,43)(50,66)(51,67)(52,68)(53,69)(54,70)(55,64)(56,65)(57,72)(58,73)(59,74)(60,75)(61,76)(62,77)(63,71)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,99)(106,122)(107,123)(108,124)(109,125)(110,126)(111,120)(112,121)(113,128)(114,129)(115,130)(116,131)(117,132)(118,133)(119,127)(134,150)(135,151)(136,152)(137,153)(138,154)(139,148)(140,149)(141,156)(142,157)(143,158)(144,159)(145,160)(146,161)(147,155)(162,178)(163,179)(164,180)(165,181)(166,182)(167,176)(168,177)(169,184)(170,185)(171,186)(172,187)(173,188)(174,189)(175,183)(190,206)(191,207)(192,208)(193,209)(194,210)(195,204)(196,205)(197,212)(198,213)(199,214)(200,215)(201,216)(202,217)(203,211), (1,35)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(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)(36,47)(37,48)(38,49)(39,43)(40,44)(41,45)(42,46)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77)(57,67)(58,68)(59,69)(60,70)(61,64)(62,65)(63,66)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105)(85,95)(86,96)(87,97)(88,98)(89,92)(90,93)(91,94)(106,127)(107,128)(108,129)(109,130)(110,131)(111,132)(112,133)(113,123)(114,124)(115,125)(116,126)(117,120)(118,121)(119,122)(134,155)(135,156)(136,157)(137,158)(138,159)(139,160)(140,161)(141,151)(142,152)(143,153)(144,154)(145,148)(146,149)(147,150)(162,183)(163,184)(164,185)(165,186)(166,187)(167,188)(168,189)(169,179)(170,180)(171,181)(172,182)(173,176)(174,177)(175,178)(190,211)(191,212)(192,213)(193,214)(194,215)(195,216)(196,217)(197,207)(198,208)(199,209)(200,210)(201,204)(202,205)(203,206), (1,150)(2,151)(3,152)(4,153)(5,154)(6,148)(7,149)(8,118)(9,119)(10,113)(11,114)(12,115)(13,116)(14,117)(15,122)(16,123)(17,124)(18,125)(19,126)(20,120)(21,121)(22,132)(23,133)(24,127)(25,128)(26,129)(27,130)(28,131)(29,141)(30,142)(31,143)(32,144)(33,145)(34,146)(35,147)(36,138)(37,139)(38,140)(39,134)(40,135)(41,136)(42,137)(43,155)(44,156)(45,157)(46,158)(47,159)(48,160)(49,161)(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), (1,35)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,224)(9,218)(10,219)(11,220)(12,221)(13,222)(14,223)(15,24)(16,25)(17,26)(18,27)(19,28)(20,22)(21,23)(36,47)(37,48)(38,49)(39,43)(40,44)(41,45)(42,46)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77)(57,67)(58,68)(59,69)(60,70)(61,64)(62,65)(63,66)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105)(85,95)(86,96)(87,97)(88,98)(89,92)(90,93)(91,94)(106,127)(107,128)(108,129)(109,130)(110,131)(111,132)(112,133)(113,123)(114,124)(115,125)(116,126)(117,120)(118,121)(119,122)(134,147)(135,141)(136,142)(137,143)(138,144)(139,145)(140,146)(148,160)(149,161)(150,155)(151,156)(152,157)(153,158)(154,159)(162,175)(163,169)(164,170)(165,171)(166,172)(167,173)(168,174)(176,188)(177,189)(178,183)(179,184)(180,185)(181,186)(182,187)(190,203)(191,197)(192,198)(193,199)(194,200)(195,201)(196,202)(204,216)(205,217)(206,211)(207,212)(208,213)(209,214)(210,215), (1,91,35,94)(2,85,29,95)(3,86,30,96)(4,87,31,97)(5,88,32,98)(6,89,33,92)(7,90,34,93)(8,177,21,174)(9,178,15,175)(10,179,16,169)(11,180,17,170)(12,181,18,171)(13,182,19,172)(14,176,20,173)(22,167,223,188)(23,168,224,189)(24,162,218,183)(25,163,219,184)(26,164,220,185)(27,165,221,186)(28,166,222,187)(36,103,47,82)(37,104,48,83)(38,105,49,84)(39,99,43,78)(40,100,44,79)(41,101,45,80)(42,102,46,81)(50,127,71,106)(51,128,72,107)(52,129,73,108)(53,130,74,109)(54,131,75,110)(55,132,76,111)(56,133,77,112)(57,123,67,113)(58,124,68,114)(59,125,69,115)(60,126,70,116)(61,120,64,117)(62,121,65,118)(63,122,66,119)(134,190,155,211)(135,191,156,212)(136,192,157,213)(137,193,158,214)(138,194,159,215)(139,195,160,216)(140,196,161,217)(141,197,151,207)(142,198,152,208)(143,199,153,209)(144,200,154,210)(145,201,148,204)(146,202,149,205)(147,203,150,206), (1,50,39,66)(2,51,40,67)(3,52,41,68)(4,53,42,69)(5,54,36,70)(6,55,37,64)(7,56,38,65)(8,202,23,217)(9,203,24,211)(10,197,25,212)(11,198,26,213)(12,199,27,214)(13,200,28,215)(14,201,22,216)(15,206,218,190)(16,207,219,191)(17,208,220,192)(18,209,221,193)(19,210,222,194)(20,204,223,195)(21,205,224,196)(29,72,44,57)(30,73,45,58)(31,74,46,59)(32,75,47,60)(33,76,48,61)(34,77,49,62)(35,71,43,63)(78,122,94,106)(79,123,95,107)(80,124,96,108)(81,125,97,109)(82,126,98,110)(83,120,92,111)(84,121,93,112)(85,128,100,113)(86,129,101,114)(87,130,102,115)(88,131,103,116)(89,132,104,117)(90,133,105,118)(91,127,99,119)(134,178,150,162)(135,179,151,163)(136,180,152,164)(137,181,153,165)(138,182,154,166)(139,176,148,167)(140,177,149,168)(141,184,156,169)(142,185,157,170)(143,186,158,171)(144,187,159,172)(145,188,160,173)(146,189,161,174)(147,183,155,175)>;
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), (1,39)(2,40)(3,41)(4,42)(5,36)(6,37)(7,38)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,22)(15,218)(16,219)(17,220)(18,221)(19,222)(20,223)(21,224)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,43)(50,66)(51,67)(52,68)(53,69)(54,70)(55,64)(56,65)(57,72)(58,73)(59,74)(60,75)(61,76)(62,77)(63,71)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,99)(106,122)(107,123)(108,124)(109,125)(110,126)(111,120)(112,121)(113,128)(114,129)(115,130)(116,131)(117,132)(118,133)(119,127)(134,150)(135,151)(136,152)(137,153)(138,154)(139,148)(140,149)(141,156)(142,157)(143,158)(144,159)(145,160)(146,161)(147,155)(162,178)(163,179)(164,180)(165,181)(166,182)(167,176)(168,177)(169,184)(170,185)(171,186)(172,187)(173,188)(174,189)(175,183)(190,206)(191,207)(192,208)(193,209)(194,210)(195,204)(196,205)(197,212)(198,213)(199,214)(200,215)(201,216)(202,217)(203,211), (1,35)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(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)(36,47)(37,48)(38,49)(39,43)(40,44)(41,45)(42,46)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77)(57,67)(58,68)(59,69)(60,70)(61,64)(62,65)(63,66)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105)(85,95)(86,96)(87,97)(88,98)(89,92)(90,93)(91,94)(106,127)(107,128)(108,129)(109,130)(110,131)(111,132)(112,133)(113,123)(114,124)(115,125)(116,126)(117,120)(118,121)(119,122)(134,155)(135,156)(136,157)(137,158)(138,159)(139,160)(140,161)(141,151)(142,152)(143,153)(144,154)(145,148)(146,149)(147,150)(162,183)(163,184)(164,185)(165,186)(166,187)(167,188)(168,189)(169,179)(170,180)(171,181)(172,182)(173,176)(174,177)(175,178)(190,211)(191,212)(192,213)(193,214)(194,215)(195,216)(196,217)(197,207)(198,208)(199,209)(200,210)(201,204)(202,205)(203,206), (1,150)(2,151)(3,152)(4,153)(5,154)(6,148)(7,149)(8,118)(9,119)(10,113)(11,114)(12,115)(13,116)(14,117)(15,122)(16,123)(17,124)(18,125)(19,126)(20,120)(21,121)(22,132)(23,133)(24,127)(25,128)(26,129)(27,130)(28,131)(29,141)(30,142)(31,143)(32,144)(33,145)(34,146)(35,147)(36,138)(37,139)(38,140)(39,134)(40,135)(41,136)(42,137)(43,155)(44,156)(45,157)(46,158)(47,159)(48,160)(49,161)(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), (1,35)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,224)(9,218)(10,219)(11,220)(12,221)(13,222)(14,223)(15,24)(16,25)(17,26)(18,27)(19,28)(20,22)(21,23)(36,47)(37,48)(38,49)(39,43)(40,44)(41,45)(42,46)(50,71)(51,72)(52,73)(53,74)(54,75)(55,76)(56,77)(57,67)(58,68)(59,69)(60,70)(61,64)(62,65)(63,66)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105)(85,95)(86,96)(87,97)(88,98)(89,92)(90,93)(91,94)(106,127)(107,128)(108,129)(109,130)(110,131)(111,132)(112,133)(113,123)(114,124)(115,125)(116,126)(117,120)(118,121)(119,122)(134,147)(135,141)(136,142)(137,143)(138,144)(139,145)(140,146)(148,160)(149,161)(150,155)(151,156)(152,157)(153,158)(154,159)(162,175)(163,169)(164,170)(165,171)(166,172)(167,173)(168,174)(176,188)(177,189)(178,183)(179,184)(180,185)(181,186)(182,187)(190,203)(191,197)(192,198)(193,199)(194,200)(195,201)(196,202)(204,216)(205,217)(206,211)(207,212)(208,213)(209,214)(210,215), (1,91,35,94)(2,85,29,95)(3,86,30,96)(4,87,31,97)(5,88,32,98)(6,89,33,92)(7,90,34,93)(8,177,21,174)(9,178,15,175)(10,179,16,169)(11,180,17,170)(12,181,18,171)(13,182,19,172)(14,176,20,173)(22,167,223,188)(23,168,224,189)(24,162,218,183)(25,163,219,184)(26,164,220,185)(27,165,221,186)(28,166,222,187)(36,103,47,82)(37,104,48,83)(38,105,49,84)(39,99,43,78)(40,100,44,79)(41,101,45,80)(42,102,46,81)(50,127,71,106)(51,128,72,107)(52,129,73,108)(53,130,74,109)(54,131,75,110)(55,132,76,111)(56,133,77,112)(57,123,67,113)(58,124,68,114)(59,125,69,115)(60,126,70,116)(61,120,64,117)(62,121,65,118)(63,122,66,119)(134,190,155,211)(135,191,156,212)(136,192,157,213)(137,193,158,214)(138,194,159,215)(139,195,160,216)(140,196,161,217)(141,197,151,207)(142,198,152,208)(143,199,153,209)(144,200,154,210)(145,201,148,204)(146,202,149,205)(147,203,150,206), (1,50,39,66)(2,51,40,67)(3,52,41,68)(4,53,42,69)(5,54,36,70)(6,55,37,64)(7,56,38,65)(8,202,23,217)(9,203,24,211)(10,197,25,212)(11,198,26,213)(12,199,27,214)(13,200,28,215)(14,201,22,216)(15,206,218,190)(16,207,219,191)(17,208,220,192)(18,209,221,193)(19,210,222,194)(20,204,223,195)(21,205,224,196)(29,72,44,57)(30,73,45,58)(31,74,46,59)(32,75,47,60)(33,76,48,61)(34,77,49,62)(35,71,43,63)(78,122,94,106)(79,123,95,107)(80,124,96,108)(81,125,97,109)(82,126,98,110)(83,120,92,111)(84,121,93,112)(85,128,100,113)(86,129,101,114)(87,130,102,115)(88,131,103,116)(89,132,104,117)(90,133,105,118)(91,127,99,119)(134,178,150,162)(135,179,151,163)(136,180,152,164)(137,181,153,165)(138,182,154,166)(139,176,148,167)(140,177,149,168)(141,184,156,169)(142,185,157,170)(143,186,158,171)(144,187,159,172)(145,188,160,173)(146,189,161,174)(147,183,155,175) );
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)], [(1,39),(2,40),(3,41),(4,42),(5,36),(6,37),(7,38),(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,22),(15,218),(16,219),(17,220),(18,221),(19,222),(20,223),(21,224),(29,44),(30,45),(31,46),(32,47),(33,48),(34,49),(35,43),(50,66),(51,67),(52,68),(53,69),(54,70),(55,64),(56,65),(57,72),(58,73),(59,74),(60,75),(61,76),(62,77),(63,71),(78,94),(79,95),(80,96),(81,97),(82,98),(83,92),(84,93),(85,100),(86,101),(87,102),(88,103),(89,104),(90,105),(91,99),(106,122),(107,123),(108,124),(109,125),(110,126),(111,120),(112,121),(113,128),(114,129),(115,130),(116,131),(117,132),(118,133),(119,127),(134,150),(135,151),(136,152),(137,153),(138,154),(139,148),(140,149),(141,156),(142,157),(143,158),(144,159),(145,160),(146,161),(147,155),(162,178),(163,179),(164,180),(165,181),(166,182),(167,176),(168,177),(169,184),(170,185),(171,186),(172,187),(173,188),(174,189),(175,183),(190,206),(191,207),(192,208),(193,209),(194,210),(195,204),(196,205),(197,212),(198,213),(199,214),(200,215),(201,216),(202,217),(203,211)], [(1,35),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(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),(36,47),(37,48),(38,49),(39,43),(40,44),(41,45),(42,46),(50,71),(51,72),(52,73),(53,74),(54,75),(55,76),(56,77),(57,67),(58,68),(59,69),(60,70),(61,64),(62,65),(63,66),(78,99),(79,100),(80,101),(81,102),(82,103),(83,104),(84,105),(85,95),(86,96),(87,97),(88,98),(89,92),(90,93),(91,94),(106,127),(107,128),(108,129),(109,130),(110,131),(111,132),(112,133),(113,123),(114,124),(115,125),(116,126),(117,120),(118,121),(119,122),(134,155),(135,156),(136,157),(137,158),(138,159),(139,160),(140,161),(141,151),(142,152),(143,153),(144,154),(145,148),(146,149),(147,150),(162,183),(163,184),(164,185),(165,186),(166,187),(167,188),(168,189),(169,179),(170,180),(171,181),(172,182),(173,176),(174,177),(175,178),(190,211),(191,212),(192,213),(193,214),(194,215),(195,216),(196,217),(197,207),(198,208),(199,209),(200,210),(201,204),(202,205),(203,206)], [(1,150),(2,151),(3,152),(4,153),(5,154),(6,148),(7,149),(8,118),(9,119),(10,113),(11,114),(12,115),(13,116),(14,117),(15,122),(16,123),(17,124),(18,125),(19,126),(20,120),(21,121),(22,132),(23,133),(24,127),(25,128),(26,129),(27,130),(28,131),(29,141),(30,142),(31,143),(32,144),(33,145),(34,146),(35,147),(36,138),(37,139),(38,140),(39,134),(40,135),(41,136),(42,137),(43,155),(44,156),(45,157),(46,158),(47,159),(48,160),(49,161),(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)], [(1,35),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,224),(9,218),(10,219),(11,220),(12,221),(13,222),(14,223),(15,24),(16,25),(17,26),(18,27),(19,28),(20,22),(21,23),(36,47),(37,48),(38,49),(39,43),(40,44),(41,45),(42,46),(50,71),(51,72),(52,73),(53,74),(54,75),(55,76),(56,77),(57,67),(58,68),(59,69),(60,70),(61,64),(62,65),(63,66),(78,99),(79,100),(80,101),(81,102),(82,103),(83,104),(84,105),(85,95),(86,96),(87,97),(88,98),(89,92),(90,93),(91,94),(106,127),(107,128),(108,129),(109,130),(110,131),(111,132),(112,133),(113,123),(114,124),(115,125),(116,126),(117,120),(118,121),(119,122),(134,147),(135,141),(136,142),(137,143),(138,144),(139,145),(140,146),(148,160),(149,161),(150,155),(151,156),(152,157),(153,158),(154,159),(162,175),(163,169),(164,170),(165,171),(166,172),(167,173),(168,174),(176,188),(177,189),(178,183),(179,184),(180,185),(181,186),(182,187),(190,203),(191,197),(192,198),(193,199),(194,200),(195,201),(196,202),(204,216),(205,217),(206,211),(207,212),(208,213),(209,214),(210,215)], [(1,91,35,94),(2,85,29,95),(3,86,30,96),(4,87,31,97),(5,88,32,98),(6,89,33,92),(7,90,34,93),(8,177,21,174),(9,178,15,175),(10,179,16,169),(11,180,17,170),(12,181,18,171),(13,182,19,172),(14,176,20,173),(22,167,223,188),(23,168,224,189),(24,162,218,183),(25,163,219,184),(26,164,220,185),(27,165,221,186),(28,166,222,187),(36,103,47,82),(37,104,48,83),(38,105,49,84),(39,99,43,78),(40,100,44,79),(41,101,45,80),(42,102,46,81),(50,127,71,106),(51,128,72,107),(52,129,73,108),(53,130,74,109),(54,131,75,110),(55,132,76,111),(56,133,77,112),(57,123,67,113),(58,124,68,114),(59,125,69,115),(60,126,70,116),(61,120,64,117),(62,121,65,118),(63,122,66,119),(134,190,155,211),(135,191,156,212),(136,192,157,213),(137,193,158,214),(138,194,159,215),(139,195,160,216),(140,196,161,217),(141,197,151,207),(142,198,152,208),(143,199,153,209),(144,200,154,210),(145,201,148,204),(146,202,149,205),(147,203,150,206)], [(1,50,39,66),(2,51,40,67),(3,52,41,68),(4,53,42,69),(5,54,36,70),(6,55,37,64),(7,56,38,65),(8,202,23,217),(9,203,24,211),(10,197,25,212),(11,198,26,213),(12,199,27,214),(13,200,28,215),(14,201,22,216),(15,206,218,190),(16,207,219,191),(17,208,220,192),(18,209,221,193),(19,210,222,194),(20,204,223,195),(21,205,224,196),(29,72,44,57),(30,73,45,58),(31,74,46,59),(32,75,47,60),(33,76,48,61),(34,77,49,62),(35,71,43,63),(78,122,94,106),(79,123,95,107),(80,124,96,108),(81,125,97,109),(82,126,98,110),(83,120,92,111),(84,121,93,112),(85,128,100,113),(86,129,101,114),(87,130,102,115),(88,131,103,116),(89,132,104,117),(90,133,105,118),(91,127,99,119),(134,178,150,162),(135,179,151,163),(136,180,152,164),(137,181,153,165),(138,182,154,166),(139,176,148,167),(140,177,149,168),(141,184,156,169),(142,185,157,170),(143,186,158,171),(144,187,159,172),(145,188,160,173),(146,189,161,174),(147,183,155,175)]])
196 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 4Q | 4R | 7A | ··· | 7F | 14A | ··· | 14R | 14S | ··· | 14AD | 14AE | ··· | 14BB | 28A | ··· | 28X | 28Y | ··· | 28CF | 28CG | ··· | 28DD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 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 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | C14 | C14 | D4 | C4○D4 | C7×D4 | C7×C4○D4 |
| kernel | C7×C22.26C24 | C2×C4×C28 | D4×C28 | C7×C4⋊D4 | C7×C4.4D4 | C7×C4⋊1D4 | C7×C4⋊Q8 | C14×C4○D4 | C22.26C24 | C2×C42 | C4×D4 | C4⋊D4 | C4.4D4 | C4⋊1D4 | C4⋊Q8 | C2×C4○D4 | C2×C28 | C28 | C2×C4 | C4 |
| # reps | 1 | 1 | 4 | 4 | 2 | 1 | 1 | 2 | 6 | 6 | 24 | 24 | 12 | 6 | 6 | 12 | 4 | 8 | 24 | 48 |
Matrix representation of C7×C22.26C24 ►in GL4(𝔽29) generated by
| 7 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 |
| 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 7 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 11 | 4 | 0 | 0 |
| 28 | 18 | 0 | 0 |
| 0 | 0 | 24 | 22 |
| 0 | 0 | 20 | 5 |
| 1 | 22 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 20 | 23 |
| 0 | 0 | 4 | 9 |
| 17 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(29))| [7,0,0,0,0,7,0,0,0,0,7,0,0,0,0,7],[28,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[11,28,0,0,4,18,0,0,0,0,24,20,0,0,22,5],[1,0,0,0,22,28,0,0,0,0,28,0,0,0,0,28],[1,0,0,0,0,1,0,0,0,0,20,4,0,0,23,9],[17,0,0,0,0,17,0,0,0,0,1,0,0,0,0,1] >;
C7×C22.26C24 in GAP, Magma, Sage, TeX
C_7\times C_2^2._{26}C_2^4 % in TeX
G:=Group("C7xC2^2.26C2^4"); // GroupNames label
G:=SmallGroup(448,1315);
// by ID
G=gap.SmallGroup(448,1315);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1597,4790,1192,416]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^7=b^2=c^2=d^2=e^2=1,f^2=c,g^2=b,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*d*e=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,f*d*f^-1=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
// generators/relations