metabelian, supersoluble, monomial
Aliases: C12.18D12, C62.114D4, C12.27(C4×S3), (C2×C12).88D6, (C3×C12).50D4, (C3×C6).16Q16, C32⋊4Q8⋊8C4, C6.23(D6⋊C4), (C3×C6).23SD16, C3⋊2(C6.SD16), C6.12(D4.S3), (C6×C12).55C22, C6.11(C3⋊Q16), C32⋊9(Q8⋊C4), C4.10(C12⋊S3), C2.2(C32⋊7Q16), C2.2(C32⋊9SD16), C2.6(C6.11D12), C22.15(C32⋊7D4), C4.2(C4×C3⋊S3), C4⋊C4.3(C3⋊S3), (C3×C4⋊C4).19S3, (C3×C12).49(C2×C4), (C32×C4⋊C4).6C2, (C2×C6).90(C3⋊D4), (C2×C32⋊4C8).7C2, (C2×C32⋊4Q8).9C2, (C3×C6).54(C22⋊C4), (C2×C4).37(C2×C3⋊S3), SmallGroup(288,285)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.114D4
G = < a,b,c,d | a6=b6=1, c4=d2=b3, ab=ba, cac-1=dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=a3c3 >
Subgroups: 420 in 126 conjugacy classes, 59 normal (21 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, C32, Dic3, C12, C12, C2×C6, C4⋊C4, C2×C8, C2×Q8, C3×C6, C3⋊C8, Dic6, C2×Dic3, C2×C12, C2×C12, Q8⋊C4, C3⋊Dic3, C3×C12, C3×C12, C62, C2×C3⋊C8, C3×C4⋊C4, C2×Dic6, C32⋊4C8, C32⋊4Q8, C32⋊4Q8, C2×C3⋊Dic3, C6×C12, C6×C12, C6.SD16, C2×C32⋊4C8, C32×C4⋊C4, C2×C32⋊4Q8, C62.114D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, SD16, Q16, C3⋊S3, C4×S3, D12, C3⋊D4, Q8⋊C4, C2×C3⋊S3, D6⋊C4, D4.S3, C3⋊Q16, C4×C3⋊S3, C12⋊S3, C32⋊7D4, C6.SD16, C6.11D12, C32⋊9SD16, C32⋊7Q16, C62.114D4
►Regular action on 288 points
Generators in S288
(1 11 285 133 141 184)(2 177 142 134 286 12)(3 13 287 135 143 178)(4 179 144 136 288 14)(5 15 281 129 137 180)(6 181 138 130 282 16)(7 9 283 131 139 182)(8 183 140 132 284 10)(17 243 40 168 203 145)(18 146 204 161 33 244)(19 245 34 162 205 147)(20 148 206 163 35 246)(21 247 36 164 207 149)(22 150 208 165 37 248)(23 241 38 166 201 151)(24 152 202 167 39 242)(25 209 159 119 199 263)(26 264 200 120 160 210)(27 211 153 113 193 257)(28 258 194 114 154 212)(29 213 155 115 195 259)(30 260 196 116 156 214)(31 215 157 117 197 261)(32 262 198 118 158 216)(41 101 106 126 250 233)(42 234 251 127 107 102)(43 103 108 128 252 235)(44 236 253 121 109 104)(45 97 110 122 254 237)(46 238 255 123 111 98)(47 99 112 124 256 239)(48 240 249 125 105 100)(49 228 62 176 89 84)(50 85 90 169 63 229)(51 230 64 170 91 86)(52 87 92 171 57 231)(53 232 58 172 93 88)(54 81 94 173 59 225)(55 226 60 174 95 82)(56 83 96 175 61 227)(65 80 270 275 218 187)(66 188 219 276 271 73)(67 74 272 277 220 189)(68 190 221 278 265 75)(69 76 266 279 222 191)(70 192 223 280 267 77)(71 78 268 273 224 185)(72 186 217 274 269 79)
(1 63 76 5 59 80)(2 73 60 6 77 64)(3 57 78 7 61 74)(4 75 62 8 79 58)(9 227 272 13 231 268)(10 269 232 14 265 228)(11 229 266 15 225 270)(12 271 226 16 267 230)(17 263 126 21 259 122)(18 123 260 22 127 264)(19 257 128 23 261 124)(20 125 262 24 121 258)(25 250 247 29 254 243)(26 244 255 30 248 251)(27 252 241 31 256 245)(28 246 249 32 242 253)(33 238 214 37 234 210)(34 211 235 38 215 239)(35 240 216 39 236 212)(36 213 237 40 209 233)(41 164 155 45 168 159)(42 160 161 46 156 165)(43 166 157 47 162 153)(44 154 163 48 158 167)(49 284 274 53 288 278)(50 279 281 54 275 285)(51 286 276 55 282 280)(52 273 283 56 277 287)(65 184 169 69 180 173)(66 174 181 70 170 177)(67 178 171 71 182 175)(68 176 183 72 172 179)(81 218 133 85 222 129)(82 130 223 86 134 219)(83 220 135 87 224 131)(84 132 217 88 136 221)(89 140 186 93 144 190)(90 191 137 94 187 141)(91 142 188 95 138 192)(92 185 139 96 189 143)(97 203 119 101 207 115)(98 116 208 102 120 204)(99 205 113 103 201 117)(100 118 202 104 114 206)(105 198 152 109 194 148)(106 149 195 110 145 199)(107 200 146 111 196 150)(108 151 197 112 147 193)
(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)(225 226 227 228 229 230 231 232)(233 234 235 236 237 238 239 240)(241 242 243 244 245 246 247 248)(249 250 251 252 253 254 255 256)(257 258 259 260 261 262 263 264)(265 266 267 268 269 270 271 272)(273 274 275 276 277 278 279 280)(281 282 283 284 285 286 287 288)
(1 24 5 20)(2 162 6 166)(3 22 7 18)(4 168 8 164)(9 244 13 248)(10 207 14 203)(11 242 15 246)(12 205 16 201)(17 132 21 136)(19 130 23 134)(25 89 29 93)(26 231 30 227)(27 95 31 91)(28 229 32 225)(33 287 37 283)(34 181 38 177)(35 285 39 281)(36 179 40 183)(41 75 45 79)(42 224 46 220)(43 73 47 77)(44 222 48 218)(49 195 53 199)(50 216 54 212)(51 193 55 197)(52 214 56 210)(57 260 61 264)(58 159 62 155)(59 258 63 262)(60 157 64 153)(65 109 69 105)(66 239 70 235)(67 107 71 111)(68 237 72 233)(74 127 78 123)(76 125 80 121)(81 154 85 158)(82 261 86 257)(83 160 87 156)(84 259 88 263)(90 118 94 114)(92 116 96 120)(97 269 101 265)(98 189 102 185)(99 267 103 271)(100 187 104 191)(106 278 110 274)(108 276 112 280)(113 226 117 230)(115 232 119 228)(122 217 126 221)(124 223 128 219)(129 163 133 167)(131 161 135 165)(137 206 141 202)(138 241 142 245)(139 204 143 208)(140 247 144 243)(145 284 149 288)(146 178 150 182)(147 282 151 286)(148 184 152 180)(169 198 173 194)(170 211 174 215)(171 196 175 200)(172 209 176 213)(186 250 190 254)(188 256 192 252)(234 273 238 277)(236 279 240 275)(249 270 253 266)(251 268 255 272)
G:=sub<Sym(288)| (1,11,285,133,141,184)(2,177,142,134,286,12)(3,13,287,135,143,178)(4,179,144,136,288,14)(5,15,281,129,137,180)(6,181,138,130,282,16)(7,9,283,131,139,182)(8,183,140,132,284,10)(17,243,40,168,203,145)(18,146,204,161,33,244)(19,245,34,162,205,147)(20,148,206,163,35,246)(21,247,36,164,207,149)(22,150,208,165,37,248)(23,241,38,166,201,151)(24,152,202,167,39,242)(25,209,159,119,199,263)(26,264,200,120,160,210)(27,211,153,113,193,257)(28,258,194,114,154,212)(29,213,155,115,195,259)(30,260,196,116,156,214)(31,215,157,117,197,261)(32,262,198,118,158,216)(41,101,106,126,250,233)(42,234,251,127,107,102)(43,103,108,128,252,235)(44,236,253,121,109,104)(45,97,110,122,254,237)(46,238,255,123,111,98)(47,99,112,124,256,239)(48,240,249,125,105,100)(49,228,62,176,89,84)(50,85,90,169,63,229)(51,230,64,170,91,86)(52,87,92,171,57,231)(53,232,58,172,93,88)(54,81,94,173,59,225)(55,226,60,174,95,82)(56,83,96,175,61,227)(65,80,270,275,218,187)(66,188,219,276,271,73)(67,74,272,277,220,189)(68,190,221,278,265,75)(69,76,266,279,222,191)(70,192,223,280,267,77)(71,78,268,273,224,185)(72,186,217,274,269,79), (1,63,76,5,59,80)(2,73,60,6,77,64)(3,57,78,7,61,74)(4,75,62,8,79,58)(9,227,272,13,231,268)(10,269,232,14,265,228)(11,229,266,15,225,270)(12,271,226,16,267,230)(17,263,126,21,259,122)(18,123,260,22,127,264)(19,257,128,23,261,124)(20,125,262,24,121,258)(25,250,247,29,254,243)(26,244,255,30,248,251)(27,252,241,31,256,245)(28,246,249,32,242,253)(33,238,214,37,234,210)(34,211,235,38,215,239)(35,240,216,39,236,212)(36,213,237,40,209,233)(41,164,155,45,168,159)(42,160,161,46,156,165)(43,166,157,47,162,153)(44,154,163,48,158,167)(49,284,274,53,288,278)(50,279,281,54,275,285)(51,286,276,55,282,280)(52,273,283,56,277,287)(65,184,169,69,180,173)(66,174,181,70,170,177)(67,178,171,71,182,175)(68,176,183,72,172,179)(81,218,133,85,222,129)(82,130,223,86,134,219)(83,220,135,87,224,131)(84,132,217,88,136,221)(89,140,186,93,144,190)(90,191,137,94,187,141)(91,142,188,95,138,192)(92,185,139,96,189,143)(97,203,119,101,207,115)(98,116,208,102,120,204)(99,205,113,103,201,117)(100,118,202,104,114,206)(105,198,152,109,194,148)(106,149,195,110,145,199)(107,200,146,111,196,150)(108,151,197,112,147,193), (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)(225,226,227,228,229,230,231,232)(233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248)(249,250,251,252,253,254,255,256)(257,258,259,260,261,262,263,264)(265,266,267,268,269,270,271,272)(273,274,275,276,277,278,279,280)(281,282,283,284,285,286,287,288), (1,24,5,20)(2,162,6,166)(3,22,7,18)(4,168,8,164)(9,244,13,248)(10,207,14,203)(11,242,15,246)(12,205,16,201)(17,132,21,136)(19,130,23,134)(25,89,29,93)(26,231,30,227)(27,95,31,91)(28,229,32,225)(33,287,37,283)(34,181,38,177)(35,285,39,281)(36,179,40,183)(41,75,45,79)(42,224,46,220)(43,73,47,77)(44,222,48,218)(49,195,53,199)(50,216,54,212)(51,193,55,197)(52,214,56,210)(57,260,61,264)(58,159,62,155)(59,258,63,262)(60,157,64,153)(65,109,69,105)(66,239,70,235)(67,107,71,111)(68,237,72,233)(74,127,78,123)(76,125,80,121)(81,154,85,158)(82,261,86,257)(83,160,87,156)(84,259,88,263)(90,118,94,114)(92,116,96,120)(97,269,101,265)(98,189,102,185)(99,267,103,271)(100,187,104,191)(106,278,110,274)(108,276,112,280)(113,226,117,230)(115,232,119,228)(122,217,126,221)(124,223,128,219)(129,163,133,167)(131,161,135,165)(137,206,141,202)(138,241,142,245)(139,204,143,208)(140,247,144,243)(145,284,149,288)(146,178,150,182)(147,282,151,286)(148,184,152,180)(169,198,173,194)(170,211,174,215)(171,196,175,200)(172,209,176,213)(186,250,190,254)(188,256,192,252)(234,273,238,277)(236,279,240,275)(249,270,253,266)(251,268,255,272)>;
G:=Group( (1,11,285,133,141,184)(2,177,142,134,286,12)(3,13,287,135,143,178)(4,179,144,136,288,14)(5,15,281,129,137,180)(6,181,138,130,282,16)(7,9,283,131,139,182)(8,183,140,132,284,10)(17,243,40,168,203,145)(18,146,204,161,33,244)(19,245,34,162,205,147)(20,148,206,163,35,246)(21,247,36,164,207,149)(22,150,208,165,37,248)(23,241,38,166,201,151)(24,152,202,167,39,242)(25,209,159,119,199,263)(26,264,200,120,160,210)(27,211,153,113,193,257)(28,258,194,114,154,212)(29,213,155,115,195,259)(30,260,196,116,156,214)(31,215,157,117,197,261)(32,262,198,118,158,216)(41,101,106,126,250,233)(42,234,251,127,107,102)(43,103,108,128,252,235)(44,236,253,121,109,104)(45,97,110,122,254,237)(46,238,255,123,111,98)(47,99,112,124,256,239)(48,240,249,125,105,100)(49,228,62,176,89,84)(50,85,90,169,63,229)(51,230,64,170,91,86)(52,87,92,171,57,231)(53,232,58,172,93,88)(54,81,94,173,59,225)(55,226,60,174,95,82)(56,83,96,175,61,227)(65,80,270,275,218,187)(66,188,219,276,271,73)(67,74,272,277,220,189)(68,190,221,278,265,75)(69,76,266,279,222,191)(70,192,223,280,267,77)(71,78,268,273,224,185)(72,186,217,274,269,79), (1,63,76,5,59,80)(2,73,60,6,77,64)(3,57,78,7,61,74)(4,75,62,8,79,58)(9,227,272,13,231,268)(10,269,232,14,265,228)(11,229,266,15,225,270)(12,271,226,16,267,230)(17,263,126,21,259,122)(18,123,260,22,127,264)(19,257,128,23,261,124)(20,125,262,24,121,258)(25,250,247,29,254,243)(26,244,255,30,248,251)(27,252,241,31,256,245)(28,246,249,32,242,253)(33,238,214,37,234,210)(34,211,235,38,215,239)(35,240,216,39,236,212)(36,213,237,40,209,233)(41,164,155,45,168,159)(42,160,161,46,156,165)(43,166,157,47,162,153)(44,154,163,48,158,167)(49,284,274,53,288,278)(50,279,281,54,275,285)(51,286,276,55,282,280)(52,273,283,56,277,287)(65,184,169,69,180,173)(66,174,181,70,170,177)(67,178,171,71,182,175)(68,176,183,72,172,179)(81,218,133,85,222,129)(82,130,223,86,134,219)(83,220,135,87,224,131)(84,132,217,88,136,221)(89,140,186,93,144,190)(90,191,137,94,187,141)(91,142,188,95,138,192)(92,185,139,96,189,143)(97,203,119,101,207,115)(98,116,208,102,120,204)(99,205,113,103,201,117)(100,118,202,104,114,206)(105,198,152,109,194,148)(106,149,195,110,145,199)(107,200,146,111,196,150)(108,151,197,112,147,193), (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)(225,226,227,228,229,230,231,232)(233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248)(249,250,251,252,253,254,255,256)(257,258,259,260,261,262,263,264)(265,266,267,268,269,270,271,272)(273,274,275,276,277,278,279,280)(281,282,283,284,285,286,287,288), (1,24,5,20)(2,162,6,166)(3,22,7,18)(4,168,8,164)(9,244,13,248)(10,207,14,203)(11,242,15,246)(12,205,16,201)(17,132,21,136)(19,130,23,134)(25,89,29,93)(26,231,30,227)(27,95,31,91)(28,229,32,225)(33,287,37,283)(34,181,38,177)(35,285,39,281)(36,179,40,183)(41,75,45,79)(42,224,46,220)(43,73,47,77)(44,222,48,218)(49,195,53,199)(50,216,54,212)(51,193,55,197)(52,214,56,210)(57,260,61,264)(58,159,62,155)(59,258,63,262)(60,157,64,153)(65,109,69,105)(66,239,70,235)(67,107,71,111)(68,237,72,233)(74,127,78,123)(76,125,80,121)(81,154,85,158)(82,261,86,257)(83,160,87,156)(84,259,88,263)(90,118,94,114)(92,116,96,120)(97,269,101,265)(98,189,102,185)(99,267,103,271)(100,187,104,191)(106,278,110,274)(108,276,112,280)(113,226,117,230)(115,232,119,228)(122,217,126,221)(124,223,128,219)(129,163,133,167)(131,161,135,165)(137,206,141,202)(138,241,142,245)(139,204,143,208)(140,247,144,243)(145,284,149,288)(146,178,150,182)(147,282,151,286)(148,184,152,180)(169,198,173,194)(170,211,174,215)(171,196,175,200)(172,209,176,213)(186,250,190,254)(188,256,192,252)(234,273,238,277)(236,279,240,275)(249,270,253,266)(251,268,255,272) );
G=PermutationGroup([[(1,11,285,133,141,184),(2,177,142,134,286,12),(3,13,287,135,143,178),(4,179,144,136,288,14),(5,15,281,129,137,180),(6,181,138,130,282,16),(7,9,283,131,139,182),(8,183,140,132,284,10),(17,243,40,168,203,145),(18,146,204,161,33,244),(19,245,34,162,205,147),(20,148,206,163,35,246),(21,247,36,164,207,149),(22,150,208,165,37,248),(23,241,38,166,201,151),(24,152,202,167,39,242),(25,209,159,119,199,263),(26,264,200,120,160,210),(27,211,153,113,193,257),(28,258,194,114,154,212),(29,213,155,115,195,259),(30,260,196,116,156,214),(31,215,157,117,197,261),(32,262,198,118,158,216),(41,101,106,126,250,233),(42,234,251,127,107,102),(43,103,108,128,252,235),(44,236,253,121,109,104),(45,97,110,122,254,237),(46,238,255,123,111,98),(47,99,112,124,256,239),(48,240,249,125,105,100),(49,228,62,176,89,84),(50,85,90,169,63,229),(51,230,64,170,91,86),(52,87,92,171,57,231),(53,232,58,172,93,88),(54,81,94,173,59,225),(55,226,60,174,95,82),(56,83,96,175,61,227),(65,80,270,275,218,187),(66,188,219,276,271,73),(67,74,272,277,220,189),(68,190,221,278,265,75),(69,76,266,279,222,191),(70,192,223,280,267,77),(71,78,268,273,224,185),(72,186,217,274,269,79)], [(1,63,76,5,59,80),(2,73,60,6,77,64),(3,57,78,7,61,74),(4,75,62,8,79,58),(9,227,272,13,231,268),(10,269,232,14,265,228),(11,229,266,15,225,270),(12,271,226,16,267,230),(17,263,126,21,259,122),(18,123,260,22,127,264),(19,257,128,23,261,124),(20,125,262,24,121,258),(25,250,247,29,254,243),(26,244,255,30,248,251),(27,252,241,31,256,245),(28,246,249,32,242,253),(33,238,214,37,234,210),(34,211,235,38,215,239),(35,240,216,39,236,212),(36,213,237,40,209,233),(41,164,155,45,168,159),(42,160,161,46,156,165),(43,166,157,47,162,153),(44,154,163,48,158,167),(49,284,274,53,288,278),(50,279,281,54,275,285),(51,286,276,55,282,280),(52,273,283,56,277,287),(65,184,169,69,180,173),(66,174,181,70,170,177),(67,178,171,71,182,175),(68,176,183,72,172,179),(81,218,133,85,222,129),(82,130,223,86,134,219),(83,220,135,87,224,131),(84,132,217,88,136,221),(89,140,186,93,144,190),(90,191,137,94,187,141),(91,142,188,95,138,192),(92,185,139,96,189,143),(97,203,119,101,207,115),(98,116,208,102,120,204),(99,205,113,103,201,117),(100,118,202,104,114,206),(105,198,152,109,194,148),(106,149,195,110,145,199),(107,200,146,111,196,150),(108,151,197,112,147,193)], [(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),(225,226,227,228,229,230,231,232),(233,234,235,236,237,238,239,240),(241,242,243,244,245,246,247,248),(249,250,251,252,253,254,255,256),(257,258,259,260,261,262,263,264),(265,266,267,268,269,270,271,272),(273,274,275,276,277,278,279,280),(281,282,283,284,285,286,287,288)], [(1,24,5,20),(2,162,6,166),(3,22,7,18),(4,168,8,164),(9,244,13,248),(10,207,14,203),(11,242,15,246),(12,205,16,201),(17,132,21,136),(19,130,23,134),(25,89,29,93),(26,231,30,227),(27,95,31,91),(28,229,32,225),(33,287,37,283),(34,181,38,177),(35,285,39,281),(36,179,40,183),(41,75,45,79),(42,224,46,220),(43,73,47,77),(44,222,48,218),(49,195,53,199),(50,216,54,212),(51,193,55,197),(52,214,56,210),(57,260,61,264),(58,159,62,155),(59,258,63,262),(60,157,64,153),(65,109,69,105),(66,239,70,235),(67,107,71,111),(68,237,72,233),(74,127,78,123),(76,125,80,121),(81,154,85,158),(82,261,86,257),(83,160,87,156),(84,259,88,263),(90,118,94,114),(92,116,96,120),(97,269,101,265),(98,189,102,185),(99,267,103,271),(100,187,104,191),(106,278,110,274),(108,276,112,280),(113,226,117,230),(115,232,119,228),(122,217,126,221),(124,223,128,219),(129,163,133,167),(131,161,135,165),(137,206,141,202),(138,241,142,245),(139,204,143,208),(140,247,144,243),(145,284,149,288),(146,178,150,182),(147,282,151,286),(148,184,152,180),(169,198,173,194),(170,211,174,215),(171,196,175,200),(172,209,176,213),(186,250,190,254),(188,256,192,252),(234,273,238,277),(236,279,240,275),(249,270,253,266),(251,268,255,272)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6L | 8A | 8B | 8C | 8D | 12A | ··· | 12X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 36 | 36 | 2 | ··· | 2 | 18 | 18 | 18 | 18 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D6 | SD16 | Q16 | C4×S3 | D12 | C3⋊D4 | D4.S3 | C3⋊Q16 |
| kernel | C62.114D4 | C2×C32⋊4C8 | C32×C4⋊C4 | C2×C32⋊4Q8 | C32⋊4Q8 | C3×C4⋊C4 | C3×C12 | C62 | C2×C12 | C3×C6 | C3×C6 | C12 | C12 | C2×C6 | C6 | C6 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 4 | 2 | 2 | 8 | 8 | 8 | 4 | 4 |
Matrix representation of C62.114D4 ►in GL6(𝔽73)
| 1 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 61 | 22 | 0 | 0 | 0 | 0 |
| 10 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 67 | 6 | 0 | 0 |
| 0 | 0 | 67 | 67 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 48 |
| 0 | 0 | 0 | 0 | 11 | 36 |
| 32 | 63 | 0 | 0 | 0 | 0 |
| 22 | 41 | 0 | 0 | 0 | 0 |
| 0 | 0 | 65 | 64 | 0 | 0 |
| 0 | 0 | 64 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 50 | 55 |
| 0 | 0 | 0 | 0 | 5 | 23 |
G:=sub<GL(6,GF(73))| [1,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,0,0,1,0],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[61,10,0,0,0,0,22,12,0,0,0,0,0,0,67,67,0,0,0,0,6,67,0,0,0,0,0,0,37,11,0,0,0,0,48,36],[32,22,0,0,0,0,63,41,0,0,0,0,0,0,65,64,0,0,0,0,64,8,0,0,0,0,0,0,50,5,0,0,0,0,55,23] >;
C62.114D4 in GAP, Magma, Sage, TeX
C_6^2._{114}D_4 % in TeX
G:=Group("C6^2.114D4"); // GroupNames label
G:=SmallGroup(288,285);
// by ID
G=gap.SmallGroup(288,285);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,141,36,675,346,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=1,c^4=d^2=b^3,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^3*c^3>;
// generators/relations