metabelian, supersoluble, monomial
Aliases: C62.117D4, (C3×Q8)⋊3Dic3, (C3×C12).58D4, (C2×C12).95D6, (C6×Q8).18S3, (C3×C6).17Q16, (Q8×C32)⋊6C4, Q8⋊2(C3⋊Dic3), (C3×C6).30SD16, C3⋊3(Q8⋊2Dic3), C12.40(C3⋊D4), (C6×C12).62C22, C12.14(C2×Dic3), C6.12(C3⋊Q16), C2.6(C62⋊5C4), C6.13(Q8⋊2S3), C32⋊13(Q8⋊C4), C4.14(C32⋊7D4), C2.3(C32⋊7Q16), C12⋊Dic3.14C2, C6.26(C6.D4), C2.3(C32⋊11SD16), C22.18(C32⋊7D4), (Q8×C3×C6).3C2, C4.2(C2×C3⋊Dic3), (C3×C12).53(C2×C4), (C2×Q8).3(C3⋊S3), (C2×C6).93(C3⋊D4), (C2×C32⋊4C8).9C2, (C3×C6).74(C22⋊C4), (C2×C4).41(C2×C3⋊S3), SmallGroup(288,310)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.117D4
G = < a,b,c,d | a6=b6=1, c4=b3, d2=a3, ab=ba, cac-1=dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=a3c3 >
Subgroups: 356 in 126 conjugacy classes, 69 normal (21 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Q8, C32, Dic3, C12, C12, C2×C6, C4⋊C4, C2×C8, C2×Q8, C3×C6, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, Q8⋊C4, C3⋊Dic3, C3×C12, C3×C12, C62, C2×C3⋊C8, C4⋊Dic3, C6×Q8, C32⋊4C8, C2×C3⋊Dic3, C6×C12, C6×C12, Q8×C32, Q8×C32, Q8⋊2Dic3, C2×C32⋊4C8, C12⋊Dic3, Q8×C3×C6, C62.117D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, SD16, Q16, C3⋊S3, C2×Dic3, C3⋊D4, Q8⋊C4, C3⋊Dic3, C2×C3⋊S3, Q8⋊2S3, C3⋊Q16, C6.D4, C2×C3⋊Dic3, C32⋊7D4, Q8⋊2Dic3, C32⋊11SD16, C32⋊7Q16, C62⋊5C4, C62.117D4
►Regular action on 288 points
Generators in S288
(1 227 285 65 109 99)(2 100 110 66 286 228)(3 229 287 67 111 101)(4 102 112 68 288 230)(5 231 281 69 105 103)(6 104 106 70 282 232)(7 225 283 71 107 97)(8 98 108 72 284 226)(9 189 117 183 59 54)(10 55 60 184 118 190)(11 191 119 177 61 56)(12 49 62 178 120 192)(13 185 113 179 63 50)(14 51 64 180 114 186)(15 187 115 181 57 52)(16 53 58 182 116 188)(17 252 31 89 124 208)(18 201 125 90 32 253)(19 254 25 91 126 202)(20 203 127 92 26 255)(21 256 27 93 128 204)(22 205 121 94 28 249)(23 250 29 95 122 206)(24 207 123 96 30 251)(33 262 153 196 168 216)(34 209 161 197 154 263)(35 264 155 198 162 210)(36 211 163 199 156 257)(37 258 157 200 164 212)(38 213 165 193 158 259)(39 260 159 194 166 214)(40 215 167 195 160 261)(41 130 143 82 241 233)(42 234 242 83 144 131)(43 132 137 84 243 235)(44 236 244 85 138 133)(45 134 139 86 245 237)(46 238 246 87 140 135)(47 136 141 88 247 239)(48 240 248 81 142 129)(73 269 280 221 173 151)(74 152 174 222 273 270)(75 271 274 223 175 145)(76 146 176 224 275 272)(77 265 276 217 169 147)(78 148 170 218 277 266)(79 267 278 219 171 149)(80 150 172 220 279 268)
(1 63 172 5 59 176)(2 169 60 6 173 64)(3 57 174 7 61 170)(4 171 62 8 175 58)(9 275 285 13 279 281)(10 282 280 14 286 276)(11 277 287 15 273 283)(12 284 274 16 288 278)(17 263 82 21 259 86)(18 87 260 22 83 264)(19 257 84 23 261 88)(20 81 262 24 85 258)(25 211 235 29 215 239)(26 240 216 30 236 212)(27 213 237 31 209 233)(28 234 210 32 238 214)(33 251 244 37 255 248)(34 241 256 38 245 252)(35 253 246 39 249 242)(36 243 250 40 247 254)(41 93 165 45 89 161)(42 162 90 46 166 94)(43 95 167 47 91 163)(44 164 92 48 168 96)(49 226 223 53 230 219)(50 220 231 54 224 227)(51 228 217 55 232 221)(52 222 225 56 218 229)(65 185 268 69 189 272)(66 265 190 70 269 186)(67 187 270 71 191 266)(68 267 192 72 271 188)(73 114 110 77 118 106)(74 107 119 78 111 115)(75 116 112 79 120 108)(76 109 113 80 105 117)(97 177 148 101 181 152)(98 145 182 102 149 178)(99 179 150 103 183 146)(100 147 184 104 151 180)(121 131 198 125 135 194)(122 195 136 126 199 132)(123 133 200 127 129 196)(124 197 130 128 193 134)(137 206 160 141 202 156)(138 157 203 142 153 207)(139 208 154 143 204 158)(140 159 205 144 155 201)
(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 92 65 20)(2 23 66 95)(3 90 67 18)(4 21 68 93)(5 96 69 24)(6 19 70 91)(7 94 71 22)(8 17 72 89)(9 153 183 216)(10 211 184 156)(11 159 177 214)(12 209 178 154)(13 157 179 212)(14 215 180 160)(15 155 181 210)(16 213 182 158)(25 104 202 282)(26 285 203 99)(27 102 204 288)(28 283 205 97)(29 100 206 286)(30 281 207 103)(31 98 208 284)(32 287 201 101)(33 54 196 117)(34 120 197 49)(35 52 198 115)(36 118 199 55)(37 50 200 113)(38 116 193 53)(39 56 194 119)(40 114 195 51)(41 171 82 267)(42 270 83 174)(43 169 84 265)(44 268 85 172)(45 175 86 271)(46 266 87 170)(47 173 88 269)(48 272 81 176)(57 162 187 264)(58 259 188 165)(59 168 189 262)(60 257 190 163)(61 166 191 260)(62 263 192 161)(63 164 185 258)(64 261 186 167)(73 136 221 247)(74 242 222 131)(75 134 223 245)(76 248 224 129)(77 132 217 243)(78 246 218 135)(79 130 219 241)(80 244 220 133)(105 251 231 123)(106 126 232 254)(107 249 225 121)(108 124 226 252)(109 255 227 127)(110 122 228 250)(111 253 229 125)(112 128 230 256)(137 276 235 147)(138 150 236 279)(139 274 237 145)(140 148 238 277)(141 280 239 151)(142 146 240 275)(143 278 233 149)(144 152 234 273)
G:=sub<Sym(288)| (1,227,285,65,109,99)(2,100,110,66,286,228)(3,229,287,67,111,101)(4,102,112,68,288,230)(5,231,281,69,105,103)(6,104,106,70,282,232)(7,225,283,71,107,97)(8,98,108,72,284,226)(9,189,117,183,59,54)(10,55,60,184,118,190)(11,191,119,177,61,56)(12,49,62,178,120,192)(13,185,113,179,63,50)(14,51,64,180,114,186)(15,187,115,181,57,52)(16,53,58,182,116,188)(17,252,31,89,124,208)(18,201,125,90,32,253)(19,254,25,91,126,202)(20,203,127,92,26,255)(21,256,27,93,128,204)(22,205,121,94,28,249)(23,250,29,95,122,206)(24,207,123,96,30,251)(33,262,153,196,168,216)(34,209,161,197,154,263)(35,264,155,198,162,210)(36,211,163,199,156,257)(37,258,157,200,164,212)(38,213,165,193,158,259)(39,260,159,194,166,214)(40,215,167,195,160,261)(41,130,143,82,241,233)(42,234,242,83,144,131)(43,132,137,84,243,235)(44,236,244,85,138,133)(45,134,139,86,245,237)(46,238,246,87,140,135)(47,136,141,88,247,239)(48,240,248,81,142,129)(73,269,280,221,173,151)(74,152,174,222,273,270)(75,271,274,223,175,145)(76,146,176,224,275,272)(77,265,276,217,169,147)(78,148,170,218,277,266)(79,267,278,219,171,149)(80,150,172,220,279,268), (1,63,172,5,59,176)(2,169,60,6,173,64)(3,57,174,7,61,170)(4,171,62,8,175,58)(9,275,285,13,279,281)(10,282,280,14,286,276)(11,277,287,15,273,283)(12,284,274,16,288,278)(17,263,82,21,259,86)(18,87,260,22,83,264)(19,257,84,23,261,88)(20,81,262,24,85,258)(25,211,235,29,215,239)(26,240,216,30,236,212)(27,213,237,31,209,233)(28,234,210,32,238,214)(33,251,244,37,255,248)(34,241,256,38,245,252)(35,253,246,39,249,242)(36,243,250,40,247,254)(41,93,165,45,89,161)(42,162,90,46,166,94)(43,95,167,47,91,163)(44,164,92,48,168,96)(49,226,223,53,230,219)(50,220,231,54,224,227)(51,228,217,55,232,221)(52,222,225,56,218,229)(65,185,268,69,189,272)(66,265,190,70,269,186)(67,187,270,71,191,266)(68,267,192,72,271,188)(73,114,110,77,118,106)(74,107,119,78,111,115)(75,116,112,79,120,108)(76,109,113,80,105,117)(97,177,148,101,181,152)(98,145,182,102,149,178)(99,179,150,103,183,146)(100,147,184,104,151,180)(121,131,198,125,135,194)(122,195,136,126,199,132)(123,133,200,127,129,196)(124,197,130,128,193,134)(137,206,160,141,202,156)(138,157,203,142,153,207)(139,208,154,143,204,158)(140,159,205,144,155,201), (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), (1,92,65,20)(2,23,66,95)(3,90,67,18)(4,21,68,93)(5,96,69,24)(6,19,70,91)(7,94,71,22)(8,17,72,89)(9,153,183,216)(10,211,184,156)(11,159,177,214)(12,209,178,154)(13,157,179,212)(14,215,180,160)(15,155,181,210)(16,213,182,158)(25,104,202,282)(26,285,203,99)(27,102,204,288)(28,283,205,97)(29,100,206,286)(30,281,207,103)(31,98,208,284)(32,287,201,101)(33,54,196,117)(34,120,197,49)(35,52,198,115)(36,118,199,55)(37,50,200,113)(38,116,193,53)(39,56,194,119)(40,114,195,51)(41,171,82,267)(42,270,83,174)(43,169,84,265)(44,268,85,172)(45,175,86,271)(46,266,87,170)(47,173,88,269)(48,272,81,176)(57,162,187,264)(58,259,188,165)(59,168,189,262)(60,257,190,163)(61,166,191,260)(62,263,192,161)(63,164,185,258)(64,261,186,167)(73,136,221,247)(74,242,222,131)(75,134,223,245)(76,248,224,129)(77,132,217,243)(78,246,218,135)(79,130,219,241)(80,244,220,133)(105,251,231,123)(106,126,232,254)(107,249,225,121)(108,124,226,252)(109,255,227,127)(110,122,228,250)(111,253,229,125)(112,128,230,256)(137,276,235,147)(138,150,236,279)(139,274,237,145)(140,148,238,277)(141,280,239,151)(142,146,240,275)(143,278,233,149)(144,152,234,273)>;
G:=Group( (1,227,285,65,109,99)(2,100,110,66,286,228)(3,229,287,67,111,101)(4,102,112,68,288,230)(5,231,281,69,105,103)(6,104,106,70,282,232)(7,225,283,71,107,97)(8,98,108,72,284,226)(9,189,117,183,59,54)(10,55,60,184,118,190)(11,191,119,177,61,56)(12,49,62,178,120,192)(13,185,113,179,63,50)(14,51,64,180,114,186)(15,187,115,181,57,52)(16,53,58,182,116,188)(17,252,31,89,124,208)(18,201,125,90,32,253)(19,254,25,91,126,202)(20,203,127,92,26,255)(21,256,27,93,128,204)(22,205,121,94,28,249)(23,250,29,95,122,206)(24,207,123,96,30,251)(33,262,153,196,168,216)(34,209,161,197,154,263)(35,264,155,198,162,210)(36,211,163,199,156,257)(37,258,157,200,164,212)(38,213,165,193,158,259)(39,260,159,194,166,214)(40,215,167,195,160,261)(41,130,143,82,241,233)(42,234,242,83,144,131)(43,132,137,84,243,235)(44,236,244,85,138,133)(45,134,139,86,245,237)(46,238,246,87,140,135)(47,136,141,88,247,239)(48,240,248,81,142,129)(73,269,280,221,173,151)(74,152,174,222,273,270)(75,271,274,223,175,145)(76,146,176,224,275,272)(77,265,276,217,169,147)(78,148,170,218,277,266)(79,267,278,219,171,149)(80,150,172,220,279,268), (1,63,172,5,59,176)(2,169,60,6,173,64)(3,57,174,7,61,170)(4,171,62,8,175,58)(9,275,285,13,279,281)(10,282,280,14,286,276)(11,277,287,15,273,283)(12,284,274,16,288,278)(17,263,82,21,259,86)(18,87,260,22,83,264)(19,257,84,23,261,88)(20,81,262,24,85,258)(25,211,235,29,215,239)(26,240,216,30,236,212)(27,213,237,31,209,233)(28,234,210,32,238,214)(33,251,244,37,255,248)(34,241,256,38,245,252)(35,253,246,39,249,242)(36,243,250,40,247,254)(41,93,165,45,89,161)(42,162,90,46,166,94)(43,95,167,47,91,163)(44,164,92,48,168,96)(49,226,223,53,230,219)(50,220,231,54,224,227)(51,228,217,55,232,221)(52,222,225,56,218,229)(65,185,268,69,189,272)(66,265,190,70,269,186)(67,187,270,71,191,266)(68,267,192,72,271,188)(73,114,110,77,118,106)(74,107,119,78,111,115)(75,116,112,79,120,108)(76,109,113,80,105,117)(97,177,148,101,181,152)(98,145,182,102,149,178)(99,179,150,103,183,146)(100,147,184,104,151,180)(121,131,198,125,135,194)(122,195,136,126,199,132)(123,133,200,127,129,196)(124,197,130,128,193,134)(137,206,160,141,202,156)(138,157,203,142,153,207)(139,208,154,143,204,158)(140,159,205,144,155,201), (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), (1,92,65,20)(2,23,66,95)(3,90,67,18)(4,21,68,93)(5,96,69,24)(6,19,70,91)(7,94,71,22)(8,17,72,89)(9,153,183,216)(10,211,184,156)(11,159,177,214)(12,209,178,154)(13,157,179,212)(14,215,180,160)(15,155,181,210)(16,213,182,158)(25,104,202,282)(26,285,203,99)(27,102,204,288)(28,283,205,97)(29,100,206,286)(30,281,207,103)(31,98,208,284)(32,287,201,101)(33,54,196,117)(34,120,197,49)(35,52,198,115)(36,118,199,55)(37,50,200,113)(38,116,193,53)(39,56,194,119)(40,114,195,51)(41,171,82,267)(42,270,83,174)(43,169,84,265)(44,268,85,172)(45,175,86,271)(46,266,87,170)(47,173,88,269)(48,272,81,176)(57,162,187,264)(58,259,188,165)(59,168,189,262)(60,257,190,163)(61,166,191,260)(62,263,192,161)(63,164,185,258)(64,261,186,167)(73,136,221,247)(74,242,222,131)(75,134,223,245)(76,248,224,129)(77,132,217,243)(78,246,218,135)(79,130,219,241)(80,244,220,133)(105,251,231,123)(106,126,232,254)(107,249,225,121)(108,124,226,252)(109,255,227,127)(110,122,228,250)(111,253,229,125)(112,128,230,256)(137,276,235,147)(138,150,236,279)(139,274,237,145)(140,148,238,277)(141,280,239,151)(142,146,240,275)(143,278,233,149)(144,152,234,273) );
G=PermutationGroup([[(1,227,285,65,109,99),(2,100,110,66,286,228),(3,229,287,67,111,101),(4,102,112,68,288,230),(5,231,281,69,105,103),(6,104,106,70,282,232),(7,225,283,71,107,97),(8,98,108,72,284,226),(9,189,117,183,59,54),(10,55,60,184,118,190),(11,191,119,177,61,56),(12,49,62,178,120,192),(13,185,113,179,63,50),(14,51,64,180,114,186),(15,187,115,181,57,52),(16,53,58,182,116,188),(17,252,31,89,124,208),(18,201,125,90,32,253),(19,254,25,91,126,202),(20,203,127,92,26,255),(21,256,27,93,128,204),(22,205,121,94,28,249),(23,250,29,95,122,206),(24,207,123,96,30,251),(33,262,153,196,168,216),(34,209,161,197,154,263),(35,264,155,198,162,210),(36,211,163,199,156,257),(37,258,157,200,164,212),(38,213,165,193,158,259),(39,260,159,194,166,214),(40,215,167,195,160,261),(41,130,143,82,241,233),(42,234,242,83,144,131),(43,132,137,84,243,235),(44,236,244,85,138,133),(45,134,139,86,245,237),(46,238,246,87,140,135),(47,136,141,88,247,239),(48,240,248,81,142,129),(73,269,280,221,173,151),(74,152,174,222,273,270),(75,271,274,223,175,145),(76,146,176,224,275,272),(77,265,276,217,169,147),(78,148,170,218,277,266),(79,267,278,219,171,149),(80,150,172,220,279,268)], [(1,63,172,5,59,176),(2,169,60,6,173,64),(3,57,174,7,61,170),(4,171,62,8,175,58),(9,275,285,13,279,281),(10,282,280,14,286,276),(11,277,287,15,273,283),(12,284,274,16,288,278),(17,263,82,21,259,86),(18,87,260,22,83,264),(19,257,84,23,261,88),(20,81,262,24,85,258),(25,211,235,29,215,239),(26,240,216,30,236,212),(27,213,237,31,209,233),(28,234,210,32,238,214),(33,251,244,37,255,248),(34,241,256,38,245,252),(35,253,246,39,249,242),(36,243,250,40,247,254),(41,93,165,45,89,161),(42,162,90,46,166,94),(43,95,167,47,91,163),(44,164,92,48,168,96),(49,226,223,53,230,219),(50,220,231,54,224,227),(51,228,217,55,232,221),(52,222,225,56,218,229),(65,185,268,69,189,272),(66,265,190,70,269,186),(67,187,270,71,191,266),(68,267,192,72,271,188),(73,114,110,77,118,106),(74,107,119,78,111,115),(75,116,112,79,120,108),(76,109,113,80,105,117),(97,177,148,101,181,152),(98,145,182,102,149,178),(99,179,150,103,183,146),(100,147,184,104,151,180),(121,131,198,125,135,194),(122,195,136,126,199,132),(123,133,200,127,129,196),(124,197,130,128,193,134),(137,206,160,141,202,156),(138,157,203,142,153,207),(139,208,154,143,204,158),(140,159,205,144,155,201)], [(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)], [(1,92,65,20),(2,23,66,95),(3,90,67,18),(4,21,68,93),(5,96,69,24),(6,19,70,91),(7,94,71,22),(8,17,72,89),(9,153,183,216),(10,211,184,156),(11,159,177,214),(12,209,178,154),(13,157,179,212),(14,215,180,160),(15,155,181,210),(16,213,182,158),(25,104,202,282),(26,285,203,99),(27,102,204,288),(28,283,205,97),(29,100,206,286),(30,281,207,103),(31,98,208,284),(32,287,201,101),(33,54,196,117),(34,120,197,49),(35,52,198,115),(36,118,199,55),(37,50,200,113),(38,116,193,53),(39,56,194,119),(40,114,195,51),(41,171,82,267),(42,270,83,174),(43,169,84,265),(44,268,85,172),(45,175,86,271),(46,266,87,170),(47,173,88,269),(48,272,81,176),(57,162,187,264),(58,259,188,165),(59,168,189,262),(60,257,190,163),(61,166,191,260),(62,263,192,161),(63,164,185,258),(64,261,186,167),(73,136,221,247),(74,242,222,131),(75,134,223,245),(76,248,224,129),(77,132,217,243),(78,246,218,135),(79,130,219,241),(80,244,220,133),(105,251,231,123),(106,126,232,254),(107,249,225,121),(108,124,226,252),(109,255,227,127),(110,122,228,250),(111,253,229,125),(112,128,230,256),(137,276,235,147),(138,150,236,279),(139,274,237,145),(140,148,238,277),(141,280,239,151),(142,146,240,275),(143,278,233,149),(144,152,234,273)]])
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 | Dic3 | SD16 | Q16 | C3⋊D4 | C3⋊D4 | Q8⋊2S3 | C3⋊Q16 |
| kernel | C62.117D4 | C2×C32⋊4C8 | C12⋊Dic3 | Q8×C3×C6 | Q8×C32 | C6×Q8 | C3×C12 | C62 | C2×C12 | C3×Q8 | C3×C6 | C3×C6 | C12 | C2×C6 | C6 | C6 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 4 | 8 | 2 | 2 | 8 | 8 | 4 | 4 |
Matrix representation of C62.117D4 ►in GL6(𝔽73)
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 3 | 0 | 0 |
| 0 | 0 | 72 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 72 | 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 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 50 | 55 | 0 | 0 | 0 | 0 |
| 5 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 47 | 40 | 0 | 0 |
| 0 | 0 | 36 | 26 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 57 |
| 0 | 0 | 0 | 0 | 16 | 16 |
| 48 | 37 | 0 | 0 | 0 | 0 |
| 62 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 33 | 0 | 0 |
| 0 | 0 | 37 | 47 | 0 | 0 |
| 0 | 0 | 0 | 0 | 62 | 43 |
| 0 | 0 | 0 | 0 | 43 | 11 |
G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,1,0,0,0,0,0,0,72,72,0,0,0,0,3,2,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[50,5,0,0,0,0,55,23,0,0,0,0,0,0,47,36,0,0,0,0,40,26,0,0,0,0,0,0,16,16,0,0,0,0,57,16],[48,62,0,0,0,0,37,25,0,0,0,0,0,0,26,37,0,0,0,0,33,47,0,0,0,0,0,0,62,43,0,0,0,0,43,11] >;
C62.117D4 in GAP, Magma, Sage, TeX
C_6^2._{117}D_4 % in TeX
G:=Group("C6^2.117D4"); // GroupNames label
G:=SmallGroup(288,310);
// by ID
G=gap.SmallGroup(288,310);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,120,675,346,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=1,c^4=b^3,d^2=a^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