direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C22.33C24, C14.1132- (1+4), C14.1552+ (1+4), (C4×D4)⋊11C14, (D4×C28)⋊40C2, C22⋊Q8⋊8C14, C4⋊D4.8C14, C42.C2⋊4C14, C42⋊2C2⋊2C14, C42.37(C2×C14), (C2×C14).359C24, (C2×C28).668C23, (C4×C28).278C22, C22.D4⋊5C14, C2.7(C7×2+ (1+4)), C2.5(C7×2- (1+4)), (D4×C14).217C22, C22.33(C23×C14), (C22×C28).38C22, C23.38(C22×C14), (C22×C14).94C23, (Q8×C14).181C22, (C14×C4⋊C4)⋊46C2, (C2×C4⋊C4)⋊19C14, C4⋊C4.68(C2×C14), C2.16(C14×C4○D4), (C7×C22⋊Q8)⋊35C2, C22.5(C7×C4○D4), (C2×D4).31(C2×C14), C14.235(C2×C4○D4), (C7×C4⋊D4).18C2, (C2×Q8).25(C2×C14), (C7×C42.C2)⋊21C2, (C7×C42⋊2C2)⋊13C2, (C2×C14).53(C4○D4), (C7×C4⋊C4).392C22, C22⋊C4.15(C2×C14), (C22×C4).59(C2×C14), (C2×C4).26(C22×C14), (C7×C22.D4)⋊24C2, (C7×C22⋊C4).85C22, SmallGroup(448,1322)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 322 in 218 conjugacy classes, 146 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C4 [×12], C22, C22 [×2], C22 [×8], C7, C2×C4 [×2], C2×C4 [×10], C2×C4 [×6], D4 [×5], Q8, C23, C23 [×2], C14 [×3], C14 [×4], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×12], C22×C4 [×3], C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C28 [×12], C2×C14, C2×C14 [×2], C2×C14 [×8], C2×C4⋊C4, C4×D4 [×2], C4⋊D4, C22⋊Q8, C22⋊Q8 [×2], C22.D4 [×4], C42.C2 [×2], C42⋊2C2 [×2], C2×C28 [×2], C2×C28 [×10], C2×C28 [×6], C7×D4 [×5], C7×Q8, C22×C14, C22×C14 [×2], C22.33C24, C4×C28 [×2], C7×C22⋊C4 [×10], C7×C4⋊C4 [×2], C7×C4⋊C4 [×12], C22×C28 [×3], C22×C28 [×2], D4×C14, D4×C14 [×2], Q8×C14, C14×C4⋊C4, D4×C28 [×2], C7×C4⋊D4, C7×C22⋊Q8, C7×C22⋊Q8 [×2], C7×C22.D4 [×4], C7×C42.C2 [×2], C7×C42⋊2C2 [×2], C7×C22.33C24
Quotients:
C1, C2 [×15], C22 [×35], C7, C23 [×15], C14 [×15], C4○D4 [×2], C24, C2×C14 [×35], C2×C4○D4, 2+ (1+4), 2- (1+4), C22×C14 [×15], C22.33C24, C7×C4○D4 [×2], C23×C14, C14×C4○D4, C7×2+ (1+4), C7×2- (1+4), C7×C22.33C24
Generators and relations
G = < a,b,c,d,e,f,g | a7=b2=c2=d2=g2=1, e2=c, f2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=gdg=bd=db, fef-1=be=eb, bf=fb, bg=gb, fdf-1=cd=dc, ce=ec, cf=fc, cg=gc, 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 38)(2 39)(3 40)(4 41)(5 42)(6 36)(7 37)(8 27)(9 28)(10 22)(11 23)(12 24)(13 25)(14 26)(15 221)(16 222)(17 223)(18 224)(19 218)(20 219)(21 220)(29 45)(30 46)(31 47)(32 48)(33 49)(34 43)(35 44)(50 74)(51 75)(52 76)(53 77)(54 71)(55 72)(56 73)(57 65)(58 66)(59 67)(60 68)(61 69)(62 70)(63 64)(78 102)(79 103)(80 104)(81 105)(82 99)(83 100)(84 101)(85 93)(86 94)(87 95)(88 96)(89 97)(90 98)(91 92)(106 130)(107 131)(108 132)(109 133)(110 127)(111 128)(112 129)(113 121)(114 122)(115 123)(116 124)(117 125)(118 126)(119 120)(134 158)(135 159)(136 160)(137 161)(138 155)(139 156)(140 157)(141 149)(142 150)(143 151)(144 152)(145 153)(146 154)(147 148)(162 186)(163 187)(164 188)(165 189)(166 183)(167 184)(168 185)(169 177)(170 178)(171 179)(172 180)(173 181)(174 182)(175 176)(190 214)(191 215)(192 216)(193 217)(194 211)(195 212)(196 213)(197 205)(198 206)(199 207)(200 208)(201 209)(202 210)(203 204)
(1 47)(2 48)(3 49)(4 43)(5 44)(6 45)(7 46)(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(22 223)(23 224)(24 218)(25 219)(26 220)(27 221)(28 222)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(50 59)(51 60)(52 61)(53 62)(54 63)(55 57)(56 58)(64 71)(65 72)(66 73)(67 74)(68 75)(69 76)(70 77)(78 87)(79 88)(80 89)(81 90)(82 91)(83 85)(84 86)(92 99)(93 100)(94 101)(95 102)(96 103)(97 104)(98 105)(106 115)(107 116)(108 117)(109 118)(110 119)(111 113)(112 114)(120 127)(121 128)(122 129)(123 130)(124 131)(125 132)(126 133)(134 143)(135 144)(136 145)(137 146)(138 147)(139 141)(140 142)(148 155)(149 156)(150 157)(151 158)(152 159)(153 160)(154 161)(162 171)(163 172)(164 173)(165 174)(166 175)(167 169)(168 170)(176 183)(177 184)(178 185)(179 186)(180 187)(181 188)(182 189)(190 199)(191 200)(192 201)(193 202)(194 203)(195 197)(196 198)(204 211)(205 212)(206 213)(207 214)(208 215)(209 216)(210 217)
(1 158)(2 159)(3 160)(4 161)(5 155)(6 156)(7 157)(8 126)(9 120)(10 121)(11 122)(12 123)(13 124)(14 125)(15 133)(16 127)(17 128)(18 129)(19 130)(20 131)(21 132)(22 113)(23 114)(24 115)(25 116)(26 117)(27 118)(28 119)(29 141)(30 142)(31 143)(32 144)(33 145)(34 146)(35 147)(36 139)(37 140)(38 134)(39 135)(40 136)(41 137)(42 138)(43 154)(44 148)(45 149)(46 150)(47 151)(48 152)(49 153)(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 95 47 102)(2 96 48 103)(3 97 49 104)(4 98 43 105)(5 92 44 99)(6 93 45 100)(7 94 46 101)(8 174 15 165)(9 175 16 166)(10 169 17 167)(11 170 18 168)(12 171 19 162)(13 172 20 163)(14 173 21 164)(22 177 223 184)(23 178 224 185)(24 179 218 186)(25 180 219 187)(26 181 220 188)(27 182 221 189)(28 176 222 183)(29 83 36 85)(30 84 37 86)(31 78 38 87)(32 79 39 88)(33 80 40 89)(34 81 41 90)(35 82 42 91)(50 115 59 106)(51 116 60 107)(52 117 61 108)(53 118 62 109)(54 119 63 110)(55 113 57 111)(56 114 58 112)(64 127 71 120)(65 128 72 121)(66 129 73 122)(67 130 74 123)(68 131 75 124)(69 132 76 125)(70 133 77 126)(134 207 143 214)(135 208 144 215)(136 209 145 216)(137 210 146 217)(138 204 147 211)(139 205 141 212)(140 206 142 213)(148 194 155 203)(149 195 156 197)(150 196 157 198)(151 190 158 199)(152 191 159 200)(153 192 160 201)(154 193 161 202)
(1 50 38 74)(2 51 39 75)(3 52 40 76)(4 53 41 77)(5 54 42 71)(6 55 36 72)(7 56 37 73)(8 193 27 217)(9 194 28 211)(10 195 22 212)(11 196 23 213)(12 190 24 214)(13 191 25 215)(14 192 26 216)(15 202 221 210)(16 203 222 204)(17 197 223 205)(18 198 224 206)(19 199 218 207)(20 200 219 208)(21 201 220 209)(29 65 45 57)(30 66 46 58)(31 67 47 59)(32 68 48 60)(33 69 49 61)(34 70 43 62)(35 64 44 63)(78 106 102 130)(79 107 103 131)(80 108 104 132)(81 109 105 133)(82 110 99 127)(83 111 100 128)(84 112 101 129)(85 113 93 121)(86 114 94 122)(87 115 95 123)(88 116 96 124)(89 117 97 125)(90 118 98 126)(91 119 92 120)(134 179 158 171)(135 180 159 172)(136 181 160 173)(137 182 161 174)(138 176 155 175)(139 177 156 169)(140 178 157 170)(141 184 149 167)(142 185 150 168)(143 186 151 162)(144 187 152 163)(145 188 153 164)(146 189 154 165)(147 183 148 166)
(8 27)(9 28)(10 22)(11 23)(12 24)(13 25)(14 26)(15 221)(16 222)(17 223)(18 224)(19 218)(20 219)(21 220)(134 158)(135 159)(136 160)(137 161)(138 155)(139 156)(140 157)(141 149)(142 150)(143 151)(144 152)(145 153)(146 154)(147 148)(162 186)(163 187)(164 188)(165 189)(166 183)(167 184)(168 185)(169 177)(170 178)(171 179)(172 180)(173 181)(174 182)(175 176)(190 214)(191 215)(192 216)(193 217)(194 211)(195 212)(196 213)(197 205)(198 206)(199 207)(200 208)(201 209)(202 210)(203 204)
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), (1,38)(2,39)(3,40)(4,41)(5,42)(6,36)(7,37)(8,27)(9,28)(10,22)(11,23)(12,24)(13,25)(14,26)(15,221)(16,222)(17,223)(18,224)(19,218)(20,219)(21,220)(29,45)(30,46)(31,47)(32,48)(33,49)(34,43)(35,44)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,65)(58,66)(59,67)(60,68)(61,69)(62,70)(63,64)(78,102)(79,103)(80,104)(81,105)(82,99)(83,100)(84,101)(85,93)(86,94)(87,95)(88,96)(89,97)(90,98)(91,92)(106,130)(107,131)(108,132)(109,133)(110,127)(111,128)(112,129)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,120)(134,158)(135,159)(136,160)(137,161)(138,155)(139,156)(140,157)(141,149)(142,150)(143,151)(144,152)(145,153)(146,154)(147,148)(162,186)(163,187)(164,188)(165,189)(166,183)(167,184)(168,185)(169,177)(170,178)(171,179)(172,180)(173,181)(174,182)(175,176)(190,214)(191,215)(192,216)(193,217)(194,211)(195,212)(196,213)(197,205)(198,206)(199,207)(200,208)(201,209)(202,210)(203,204), (1,47)(2,48)(3,49)(4,43)(5,44)(6,45)(7,46)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(22,223)(23,224)(24,218)(25,219)(26,220)(27,221)(28,222)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(50,59)(51,60)(52,61)(53,62)(54,63)(55,57)(56,58)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77)(78,87)(79,88)(80,89)(81,90)(82,91)(83,85)(84,86)(92,99)(93,100)(94,101)(95,102)(96,103)(97,104)(98,105)(106,115)(107,116)(108,117)(109,118)(110,119)(111,113)(112,114)(120,127)(121,128)(122,129)(123,130)(124,131)(125,132)(126,133)(134,143)(135,144)(136,145)(137,146)(138,147)(139,141)(140,142)(148,155)(149,156)(150,157)(151,158)(152,159)(153,160)(154,161)(162,171)(163,172)(164,173)(165,174)(166,175)(167,169)(168,170)(176,183)(177,184)(178,185)(179,186)(180,187)(181,188)(182,189)(190,199)(191,200)(192,201)(193,202)(194,203)(195,197)(196,198)(204,211)(205,212)(206,213)(207,214)(208,215)(209,216)(210,217), (1,158)(2,159)(3,160)(4,161)(5,155)(6,156)(7,157)(8,126)(9,120)(10,121)(11,122)(12,123)(13,124)(14,125)(15,133)(16,127)(17,128)(18,129)(19,130)(20,131)(21,132)(22,113)(23,114)(24,115)(25,116)(26,117)(27,118)(28,119)(29,141)(30,142)(31,143)(32,144)(33,145)(34,146)(35,147)(36,139)(37,140)(38,134)(39,135)(40,136)(41,137)(42,138)(43,154)(44,148)(45,149)(46,150)(47,151)(48,152)(49,153)(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), (1,95,47,102)(2,96,48,103)(3,97,49,104)(4,98,43,105)(5,92,44,99)(6,93,45,100)(7,94,46,101)(8,174,15,165)(9,175,16,166)(10,169,17,167)(11,170,18,168)(12,171,19,162)(13,172,20,163)(14,173,21,164)(22,177,223,184)(23,178,224,185)(24,179,218,186)(25,180,219,187)(26,181,220,188)(27,182,221,189)(28,176,222,183)(29,83,36,85)(30,84,37,86)(31,78,38,87)(32,79,39,88)(33,80,40,89)(34,81,41,90)(35,82,42,91)(50,115,59,106)(51,116,60,107)(52,117,61,108)(53,118,62,109)(54,119,63,110)(55,113,57,111)(56,114,58,112)(64,127,71,120)(65,128,72,121)(66,129,73,122)(67,130,74,123)(68,131,75,124)(69,132,76,125)(70,133,77,126)(134,207,143,214)(135,208,144,215)(136,209,145,216)(137,210,146,217)(138,204,147,211)(139,205,141,212)(140,206,142,213)(148,194,155,203)(149,195,156,197)(150,196,157,198)(151,190,158,199)(152,191,159,200)(153,192,160,201)(154,193,161,202), (1,50,38,74)(2,51,39,75)(3,52,40,76)(4,53,41,77)(5,54,42,71)(6,55,36,72)(7,56,37,73)(8,193,27,217)(9,194,28,211)(10,195,22,212)(11,196,23,213)(12,190,24,214)(13,191,25,215)(14,192,26,216)(15,202,221,210)(16,203,222,204)(17,197,223,205)(18,198,224,206)(19,199,218,207)(20,200,219,208)(21,201,220,209)(29,65,45,57)(30,66,46,58)(31,67,47,59)(32,68,48,60)(33,69,49,61)(34,70,43,62)(35,64,44,63)(78,106,102,130)(79,107,103,131)(80,108,104,132)(81,109,105,133)(82,110,99,127)(83,111,100,128)(84,112,101,129)(85,113,93,121)(86,114,94,122)(87,115,95,123)(88,116,96,124)(89,117,97,125)(90,118,98,126)(91,119,92,120)(134,179,158,171)(135,180,159,172)(136,181,160,173)(137,182,161,174)(138,176,155,175)(139,177,156,169)(140,178,157,170)(141,184,149,167)(142,185,150,168)(143,186,151,162)(144,187,152,163)(145,188,153,164)(146,189,154,165)(147,183,148,166), (8,27)(9,28)(10,22)(11,23)(12,24)(13,25)(14,26)(15,221)(16,222)(17,223)(18,224)(19,218)(20,219)(21,220)(134,158)(135,159)(136,160)(137,161)(138,155)(139,156)(140,157)(141,149)(142,150)(143,151)(144,152)(145,153)(146,154)(147,148)(162,186)(163,187)(164,188)(165,189)(166,183)(167,184)(168,185)(169,177)(170,178)(171,179)(172,180)(173,181)(174,182)(175,176)(190,214)(191,215)(192,216)(193,217)(194,211)(195,212)(196,213)(197,205)(198,206)(199,207)(200,208)(201,209)(202,210)(203,204)>;
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), (1,38)(2,39)(3,40)(4,41)(5,42)(6,36)(7,37)(8,27)(9,28)(10,22)(11,23)(12,24)(13,25)(14,26)(15,221)(16,222)(17,223)(18,224)(19,218)(20,219)(21,220)(29,45)(30,46)(31,47)(32,48)(33,49)(34,43)(35,44)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,65)(58,66)(59,67)(60,68)(61,69)(62,70)(63,64)(78,102)(79,103)(80,104)(81,105)(82,99)(83,100)(84,101)(85,93)(86,94)(87,95)(88,96)(89,97)(90,98)(91,92)(106,130)(107,131)(108,132)(109,133)(110,127)(111,128)(112,129)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,120)(134,158)(135,159)(136,160)(137,161)(138,155)(139,156)(140,157)(141,149)(142,150)(143,151)(144,152)(145,153)(146,154)(147,148)(162,186)(163,187)(164,188)(165,189)(166,183)(167,184)(168,185)(169,177)(170,178)(171,179)(172,180)(173,181)(174,182)(175,176)(190,214)(191,215)(192,216)(193,217)(194,211)(195,212)(196,213)(197,205)(198,206)(199,207)(200,208)(201,209)(202,210)(203,204), (1,47)(2,48)(3,49)(4,43)(5,44)(6,45)(7,46)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(22,223)(23,224)(24,218)(25,219)(26,220)(27,221)(28,222)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(50,59)(51,60)(52,61)(53,62)(54,63)(55,57)(56,58)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77)(78,87)(79,88)(80,89)(81,90)(82,91)(83,85)(84,86)(92,99)(93,100)(94,101)(95,102)(96,103)(97,104)(98,105)(106,115)(107,116)(108,117)(109,118)(110,119)(111,113)(112,114)(120,127)(121,128)(122,129)(123,130)(124,131)(125,132)(126,133)(134,143)(135,144)(136,145)(137,146)(138,147)(139,141)(140,142)(148,155)(149,156)(150,157)(151,158)(152,159)(153,160)(154,161)(162,171)(163,172)(164,173)(165,174)(166,175)(167,169)(168,170)(176,183)(177,184)(178,185)(179,186)(180,187)(181,188)(182,189)(190,199)(191,200)(192,201)(193,202)(194,203)(195,197)(196,198)(204,211)(205,212)(206,213)(207,214)(208,215)(209,216)(210,217), (1,158)(2,159)(3,160)(4,161)(5,155)(6,156)(7,157)(8,126)(9,120)(10,121)(11,122)(12,123)(13,124)(14,125)(15,133)(16,127)(17,128)(18,129)(19,130)(20,131)(21,132)(22,113)(23,114)(24,115)(25,116)(26,117)(27,118)(28,119)(29,141)(30,142)(31,143)(32,144)(33,145)(34,146)(35,147)(36,139)(37,140)(38,134)(39,135)(40,136)(41,137)(42,138)(43,154)(44,148)(45,149)(46,150)(47,151)(48,152)(49,153)(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), (1,95,47,102)(2,96,48,103)(3,97,49,104)(4,98,43,105)(5,92,44,99)(6,93,45,100)(7,94,46,101)(8,174,15,165)(9,175,16,166)(10,169,17,167)(11,170,18,168)(12,171,19,162)(13,172,20,163)(14,173,21,164)(22,177,223,184)(23,178,224,185)(24,179,218,186)(25,180,219,187)(26,181,220,188)(27,182,221,189)(28,176,222,183)(29,83,36,85)(30,84,37,86)(31,78,38,87)(32,79,39,88)(33,80,40,89)(34,81,41,90)(35,82,42,91)(50,115,59,106)(51,116,60,107)(52,117,61,108)(53,118,62,109)(54,119,63,110)(55,113,57,111)(56,114,58,112)(64,127,71,120)(65,128,72,121)(66,129,73,122)(67,130,74,123)(68,131,75,124)(69,132,76,125)(70,133,77,126)(134,207,143,214)(135,208,144,215)(136,209,145,216)(137,210,146,217)(138,204,147,211)(139,205,141,212)(140,206,142,213)(148,194,155,203)(149,195,156,197)(150,196,157,198)(151,190,158,199)(152,191,159,200)(153,192,160,201)(154,193,161,202), (1,50,38,74)(2,51,39,75)(3,52,40,76)(4,53,41,77)(5,54,42,71)(6,55,36,72)(7,56,37,73)(8,193,27,217)(9,194,28,211)(10,195,22,212)(11,196,23,213)(12,190,24,214)(13,191,25,215)(14,192,26,216)(15,202,221,210)(16,203,222,204)(17,197,223,205)(18,198,224,206)(19,199,218,207)(20,200,219,208)(21,201,220,209)(29,65,45,57)(30,66,46,58)(31,67,47,59)(32,68,48,60)(33,69,49,61)(34,70,43,62)(35,64,44,63)(78,106,102,130)(79,107,103,131)(80,108,104,132)(81,109,105,133)(82,110,99,127)(83,111,100,128)(84,112,101,129)(85,113,93,121)(86,114,94,122)(87,115,95,123)(88,116,96,124)(89,117,97,125)(90,118,98,126)(91,119,92,120)(134,179,158,171)(135,180,159,172)(136,181,160,173)(137,182,161,174)(138,176,155,175)(139,177,156,169)(140,178,157,170)(141,184,149,167)(142,185,150,168)(143,186,151,162)(144,187,152,163)(145,188,153,164)(146,189,154,165)(147,183,148,166), (8,27)(9,28)(10,22)(11,23)(12,24)(13,25)(14,26)(15,221)(16,222)(17,223)(18,224)(19,218)(20,219)(21,220)(134,158)(135,159)(136,160)(137,161)(138,155)(139,156)(140,157)(141,149)(142,150)(143,151)(144,152)(145,153)(146,154)(147,148)(162,186)(163,187)(164,188)(165,189)(166,183)(167,184)(168,185)(169,177)(170,178)(171,179)(172,180)(173,181)(174,182)(175,176)(190,214)(191,215)(192,216)(193,217)(194,211)(195,212)(196,213)(197,205)(198,206)(199,207)(200,208)(201,209)(202,210)(203,204) );
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)], [(1,38),(2,39),(3,40),(4,41),(5,42),(6,36),(7,37),(8,27),(9,28),(10,22),(11,23),(12,24),(13,25),(14,26),(15,221),(16,222),(17,223),(18,224),(19,218),(20,219),(21,220),(29,45),(30,46),(31,47),(32,48),(33,49),(34,43),(35,44),(50,74),(51,75),(52,76),(53,77),(54,71),(55,72),(56,73),(57,65),(58,66),(59,67),(60,68),(61,69),(62,70),(63,64),(78,102),(79,103),(80,104),(81,105),(82,99),(83,100),(84,101),(85,93),(86,94),(87,95),(88,96),(89,97),(90,98),(91,92),(106,130),(107,131),(108,132),(109,133),(110,127),(111,128),(112,129),(113,121),(114,122),(115,123),(116,124),(117,125),(118,126),(119,120),(134,158),(135,159),(136,160),(137,161),(138,155),(139,156),(140,157),(141,149),(142,150),(143,151),(144,152),(145,153),(146,154),(147,148),(162,186),(163,187),(164,188),(165,189),(166,183),(167,184),(168,185),(169,177),(170,178),(171,179),(172,180),(173,181),(174,182),(175,176),(190,214),(191,215),(192,216),(193,217),(194,211),(195,212),(196,213),(197,205),(198,206),(199,207),(200,208),(201,209),(202,210),(203,204)], [(1,47),(2,48),(3,49),(4,43),(5,44),(6,45),(7,46),(8,15),(9,16),(10,17),(11,18),(12,19),(13,20),(14,21),(22,223),(23,224),(24,218),(25,219),(26,220),(27,221),(28,222),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(50,59),(51,60),(52,61),(53,62),(54,63),(55,57),(56,58),(64,71),(65,72),(66,73),(67,74),(68,75),(69,76),(70,77),(78,87),(79,88),(80,89),(81,90),(82,91),(83,85),(84,86),(92,99),(93,100),(94,101),(95,102),(96,103),(97,104),(98,105),(106,115),(107,116),(108,117),(109,118),(110,119),(111,113),(112,114),(120,127),(121,128),(122,129),(123,130),(124,131),(125,132),(126,133),(134,143),(135,144),(136,145),(137,146),(138,147),(139,141),(140,142),(148,155),(149,156),(150,157),(151,158),(152,159),(153,160),(154,161),(162,171),(163,172),(164,173),(165,174),(166,175),(167,169),(168,170),(176,183),(177,184),(178,185),(179,186),(180,187),(181,188),(182,189),(190,199),(191,200),(192,201),(193,202),(194,203),(195,197),(196,198),(204,211),(205,212),(206,213),(207,214),(208,215),(209,216),(210,217)], [(1,158),(2,159),(3,160),(4,161),(5,155),(6,156),(7,157),(8,126),(9,120),(10,121),(11,122),(12,123),(13,124),(14,125),(15,133),(16,127),(17,128),(18,129),(19,130),(20,131),(21,132),(22,113),(23,114),(24,115),(25,116),(26,117),(27,118),(28,119),(29,141),(30,142),(31,143),(32,144),(33,145),(34,146),(35,147),(36,139),(37,140),(38,134),(39,135),(40,136),(41,137),(42,138),(43,154),(44,148),(45,149),(46,150),(47,151),(48,152),(49,153),(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)], [(1,95,47,102),(2,96,48,103),(3,97,49,104),(4,98,43,105),(5,92,44,99),(6,93,45,100),(7,94,46,101),(8,174,15,165),(9,175,16,166),(10,169,17,167),(11,170,18,168),(12,171,19,162),(13,172,20,163),(14,173,21,164),(22,177,223,184),(23,178,224,185),(24,179,218,186),(25,180,219,187),(26,181,220,188),(27,182,221,189),(28,176,222,183),(29,83,36,85),(30,84,37,86),(31,78,38,87),(32,79,39,88),(33,80,40,89),(34,81,41,90),(35,82,42,91),(50,115,59,106),(51,116,60,107),(52,117,61,108),(53,118,62,109),(54,119,63,110),(55,113,57,111),(56,114,58,112),(64,127,71,120),(65,128,72,121),(66,129,73,122),(67,130,74,123),(68,131,75,124),(69,132,76,125),(70,133,77,126),(134,207,143,214),(135,208,144,215),(136,209,145,216),(137,210,146,217),(138,204,147,211),(139,205,141,212),(140,206,142,213),(148,194,155,203),(149,195,156,197),(150,196,157,198),(151,190,158,199),(152,191,159,200),(153,192,160,201),(154,193,161,202)], [(1,50,38,74),(2,51,39,75),(3,52,40,76),(4,53,41,77),(5,54,42,71),(6,55,36,72),(7,56,37,73),(8,193,27,217),(9,194,28,211),(10,195,22,212),(11,196,23,213),(12,190,24,214),(13,191,25,215),(14,192,26,216),(15,202,221,210),(16,203,222,204),(17,197,223,205),(18,198,224,206),(19,199,218,207),(20,200,219,208),(21,201,220,209),(29,65,45,57),(30,66,46,58),(31,67,47,59),(32,68,48,60),(33,69,49,61),(34,70,43,62),(35,64,44,63),(78,106,102,130),(79,107,103,131),(80,108,104,132),(81,109,105,133),(82,110,99,127),(83,111,100,128),(84,112,101,129),(85,113,93,121),(86,114,94,122),(87,115,95,123),(88,116,96,124),(89,117,97,125),(90,118,98,126),(91,119,92,120),(134,179,158,171),(135,180,159,172),(136,181,160,173),(137,182,161,174),(138,176,155,175),(139,177,156,169),(140,178,157,170),(141,184,149,167),(142,185,150,168),(143,186,151,162),(144,187,152,163),(145,188,153,164),(146,189,154,165),(147,183,148,166)], [(8,27),(9,28),(10,22),(11,23),(12,24),(13,25),(14,26),(15,221),(16,222),(17,223),(18,224),(19,218),(20,219),(21,220),(134,158),(135,159),(136,160),(137,161),(138,155),(139,156),(140,157),(141,149),(142,150),(143,151),(144,152),(145,153),(146,154),(147,148),(162,186),(163,187),(164,188),(165,189),(166,183),(167,184),(168,185),(169,177),(170,178),(171,179),(172,180),(173,181),(174,182),(175,176),(190,214),(191,215),(192,216),(193,217),(194,211),(195,212),(196,213),(197,205),(198,206),(199,207),(200,208),(201,209),(202,210),(203,204)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 0 | 0 | 0 |
| 0 | 0 | 0 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 0 |
| 0 | 0 | 0 | 0 | 0 | 24 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 17 | 1 | 8 |
| 0 | 0 | 0 | 10 | 0 | 6 |
| 0 | 0 | 1 | 20 | 0 | 7 |
| 0 | 0 | 0 | 27 | 0 | 19 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 23 | 9 | 21 | 23 |
| 0 | 0 | 9 | 6 | 26 | 5 |
| 0 | 0 | 0 | 0 | 0 | 21 |
| 0 | 0 | 0 | 0 | 18 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 12 | 9 |
| 0 | 0 | 28 | 0 | 19 | 22 |
| 0 | 0 | 0 | 0 | 9 | 17 |
| 0 | 0 | 0 | 0 | 2 | 20 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 17 |
| 0 | 0 | 0 | 1 | 0 | 10 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,24,0,0,0,0,0,0,24,0,0,0,0,0,0,24,0,0,0,0,0,0,24],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,28,0,0,0,0,28,0,0,0,0,0,0,0,0,0,1,0,0,0,17,10,20,27,0,0,1,0,0,0,0,0,8,6,7,19],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,23,9,0,0,0,0,9,6,0,0,0,0,21,26,0,18,0,0,23,5,21,0],[1,0,0,0,0,0,0,28,0,0,0,0,0,0,0,28,0,0,0,0,1,0,0,0,0,0,12,19,9,2,0,0,9,22,17,20],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,17,10,0,28] >;
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 7A | ··· | 7F | 14A | ··· | 14R | 14S | ··· | 14AD | 14AE | ··· | 14AP | 28A | ··· | 28X | 28Y | ··· | 28CF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | C14 | C14 | C4○D4 | C7×C4○D4 | 2+ (1+4) | 2- (1+4) | C7×2+ (1+4) | C7×2- (1+4) |
| kernel | C7×C22.33C24 | C14×C4⋊C4 | D4×C28 | C7×C4⋊D4 | C7×C22⋊Q8 | C7×C22.D4 | C7×C42.C2 | C7×C42⋊2C2 | C22.33C24 | C2×C4⋊C4 | C4×D4 | C4⋊D4 | C22⋊Q8 | C22.D4 | C42.C2 | C42⋊2C2 | C2×C14 | C22 | C14 | C14 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 3 | 4 | 2 | 2 | 6 | 6 | 12 | 6 | 18 | 24 | 12 | 12 | 4 | 24 | 1 | 1 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_7\times C_2^2._{33}C_2^4 % in TeX
G:=Group("C7xC2^2.33C2^4"); // GroupNames label
G:=SmallGroup(448,1322);
// by ID
G=gap.SmallGroup(448,1322);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1597,1576,4790,1227,604,3363]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^7=b^2=c^2=d^2=g^2=1,e^2=c,f^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^-1=g*d*g=b*d=d*b,f*e*f^-1=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,e*g=g*e,f*g=g*f>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀