direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C12.3Q8, C4⋊C4.260D6, C12.66(C2×Q8), (C2×C12).30Q8, C6.8(C22×Q8), (C2×C6).43C24, C6⋊3(C42.C2), (C2×C4).37Dic6, C4.31(C2×Dic6), (C22×C4).374D6, (C2×C12).134C23, C22.81(S3×C23), C2.10(C22×Dic6), C22.37(C2×Dic6), C4⋊Dic3.357C22, C23.333(C22×S3), (C22×C12).73C22, (C22×C6).392C23, C22.72(D4⋊2S3), Dic3⋊C4.104C22, C22.32(Q8⋊3S3), (C4×Dic3).248C22, (C2×Dic3).183C23, (C22×Dic3).208C22, (C6×C4⋊C4).19C2, (C2×C4⋊C4).26S3, C3⋊3(C2×C42.C2), C6.70(C2×C4○D4), (C2×C6).51(C2×Q8), C2.5(C2×Q8⋊3S3), (C2×C4×Dic3).13C2, C2.14(C2×D4⋊2S3), (C2×C4⋊Dic3).43C2, (C2×C6).170(C4○D4), (C3×C4⋊C4).292C22, (C2×Dic3⋊C4).22C2, (C2×C4).139(C22×S3), SmallGroup(192,1058)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C12.3Q8
G = < a,b,c,d | a2=b12=c4=1, d2=c2, ab=ba, ac=ca, ad=da, cbc-1=b7, dbd-1=b5, dcd-1=b6c-1 >
Subgroups: 440 in 226 conjugacy classes, 127 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C4 [×12], C22, C22 [×6], C6 [×3], C6 [×4], C2×C4 [×10], C2×C4 [×20], C23, Dic3 [×8], C12 [×4], C12 [×4], C2×C6, C2×C6 [×6], C42 [×4], C4⋊C4 [×4], C4⋊C4 [×20], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Dic3 [×8], C2×Dic3 [×8], C2×C12 [×10], C2×C12 [×4], C22×C6, C2×C42, C2×C4⋊C4, C2×C4⋊C4 [×5], C42.C2 [×8], C4×Dic3 [×4], Dic3⋊C4 [×8], C4⋊Dic3 [×12], C3×C4⋊C4 [×4], C22×Dic3 [×2], C22×Dic3 [×2], C22×C12, C22×C12 [×2], C2×C42.C2, C12.3Q8 [×8], C2×C4×Dic3, C2×Dic3⋊C4 [×2], C2×C4⋊Dic3, C2×C4⋊Dic3 [×2], C6×C4⋊C4, C2×C12.3Q8
Quotients: C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D6 [×7], C2×Q8 [×6], C4○D4 [×4], C24, Dic6 [×4], C22×S3 [×7], C42.C2 [×4], C22×Q8, C2×C4○D4 [×2], C2×Dic6 [×6], D4⋊2S3 [×2], Q8⋊3S3 [×2], S3×C23, C2×C42.C2, C12.3Q8 [×4], C22×Dic6, C2×D4⋊2S3, C2×Q8⋊3S3, C2×C12.3Q8
►Regular action on 192 points
Generators in S192
(1 85)(2 86)(3 87)(4 88)(5 89)(6 90)(7 91)(8 92)(9 93)(10 94)(11 95)(12 96)(13 74)(14 75)(15 76)(16 77)(17 78)(18 79)(19 80)(20 81)(21 82)(22 83)(23 84)(24 73)(25 145)(26 146)(27 147)(28 148)(29 149)(30 150)(31 151)(32 152)(33 153)(34 154)(35 155)(36 156)(37 68)(38 69)(39 70)(40 71)(41 72)(42 61)(43 62)(44 63)(45 64)(46 65)(47 66)(48 67)(49 104)(50 105)(51 106)(52 107)(53 108)(54 97)(55 98)(56 99)(57 100)(58 101)(59 102)(60 103)(109 163)(110 164)(111 165)(112 166)(113 167)(114 168)(115 157)(116 158)(117 159)(118 160)(119 161)(120 162)(121 175)(122 176)(123 177)(124 178)(125 179)(126 180)(127 169)(128 170)(129 171)(130 172)(131 173)(132 174)(133 187)(134 188)(135 189)(136 190)(137 191)(138 192)(139 181)(140 182)(141 183)(142 184)(143 185)(144 186)
(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)
(1 102 15 68)(2 97 16 63)(3 104 17 70)(4 99 18 65)(5 106 19 72)(6 101 20 67)(7 108 21 62)(8 103 22 69)(9 98 23 64)(10 105 24 71)(11 100 13 66)(12 107 14 61)(25 140 166 131)(26 135 167 126)(27 142 168 121)(28 137 157 128)(29 144 158 123)(30 139 159 130)(31 134 160 125)(32 141 161 132)(33 136 162 127)(34 143 163 122)(35 138 164 129)(36 133 165 124)(37 85 59 76)(38 92 60 83)(39 87 49 78)(40 94 50 73)(41 89 51 80)(42 96 52 75)(43 91 53 82)(44 86 54 77)(45 93 55 84)(46 88 56 79)(47 95 57 74)(48 90 58 81)(109 176 154 185)(110 171 155 192)(111 178 156 187)(112 173 145 182)(113 180 146 189)(114 175 147 184)(115 170 148 191)(116 177 149 186)(117 172 150 181)(118 179 151 188)(119 174 152 183)(120 169 153 190)
(1 128 15 137)(2 121 16 142)(3 126 17 135)(4 131 18 140)(5 124 19 133)(6 129 20 138)(7 122 21 143)(8 127 22 136)(9 132 23 141)(10 125 24 134)(11 130 13 139)(12 123 14 144)(25 105 166 71)(26 98 167 64)(27 103 168 69)(28 108 157 62)(29 101 158 67)(30 106 159 72)(31 99 160 65)(32 104 161 70)(33 97 162 63)(34 102 163 68)(35 107 164 61)(36 100 165 66)(37 154 59 109)(38 147 60 114)(39 152 49 119)(40 145 50 112)(41 150 51 117)(42 155 52 110)(43 148 53 115)(44 153 54 120)(45 146 55 113)(46 151 56 118)(47 156 57 111)(48 149 58 116)(73 188 94 179)(74 181 95 172)(75 186 96 177)(76 191 85 170)(77 184 86 175)(78 189 87 180)(79 182 88 173)(80 187 89 178)(81 192 90 171)(82 185 91 176)(83 190 92 169)(84 183 93 174)
G:=sub<Sym(192)| (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,73)(25,145)(26,146)(27,147)(28,148)(29,149)(30,150)(31,151)(32,152)(33,153)(34,154)(35,155)(36,156)(37,68)(38,69)(39,70)(40,71)(41,72)(42,61)(43,62)(44,63)(45,64)(46,65)(47,66)(48,67)(49,104)(50,105)(51,106)(52,107)(53,108)(54,97)(55,98)(56,99)(57,100)(58,101)(59,102)(60,103)(109,163)(110,164)(111,165)(112,166)(113,167)(114,168)(115,157)(116,158)(117,159)(118,160)(119,161)(120,162)(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)(127,169)(128,170)(129,171)(130,172)(131,173)(132,174)(133,187)(134,188)(135,189)(136,190)(137,191)(138,192)(139,181)(140,182)(141,183)(142,184)(143,185)(144,186), (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), (1,102,15,68)(2,97,16,63)(3,104,17,70)(4,99,18,65)(5,106,19,72)(6,101,20,67)(7,108,21,62)(8,103,22,69)(9,98,23,64)(10,105,24,71)(11,100,13,66)(12,107,14,61)(25,140,166,131)(26,135,167,126)(27,142,168,121)(28,137,157,128)(29,144,158,123)(30,139,159,130)(31,134,160,125)(32,141,161,132)(33,136,162,127)(34,143,163,122)(35,138,164,129)(36,133,165,124)(37,85,59,76)(38,92,60,83)(39,87,49,78)(40,94,50,73)(41,89,51,80)(42,96,52,75)(43,91,53,82)(44,86,54,77)(45,93,55,84)(46,88,56,79)(47,95,57,74)(48,90,58,81)(109,176,154,185)(110,171,155,192)(111,178,156,187)(112,173,145,182)(113,180,146,189)(114,175,147,184)(115,170,148,191)(116,177,149,186)(117,172,150,181)(118,179,151,188)(119,174,152,183)(120,169,153,190), (1,128,15,137)(2,121,16,142)(3,126,17,135)(4,131,18,140)(5,124,19,133)(6,129,20,138)(7,122,21,143)(8,127,22,136)(9,132,23,141)(10,125,24,134)(11,130,13,139)(12,123,14,144)(25,105,166,71)(26,98,167,64)(27,103,168,69)(28,108,157,62)(29,101,158,67)(30,106,159,72)(31,99,160,65)(32,104,161,70)(33,97,162,63)(34,102,163,68)(35,107,164,61)(36,100,165,66)(37,154,59,109)(38,147,60,114)(39,152,49,119)(40,145,50,112)(41,150,51,117)(42,155,52,110)(43,148,53,115)(44,153,54,120)(45,146,55,113)(46,151,56,118)(47,156,57,111)(48,149,58,116)(73,188,94,179)(74,181,95,172)(75,186,96,177)(76,191,85,170)(77,184,86,175)(78,189,87,180)(79,182,88,173)(80,187,89,178)(81,192,90,171)(82,185,91,176)(83,190,92,169)(84,183,93,174)>;
G:=Group( (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,73)(25,145)(26,146)(27,147)(28,148)(29,149)(30,150)(31,151)(32,152)(33,153)(34,154)(35,155)(36,156)(37,68)(38,69)(39,70)(40,71)(41,72)(42,61)(43,62)(44,63)(45,64)(46,65)(47,66)(48,67)(49,104)(50,105)(51,106)(52,107)(53,108)(54,97)(55,98)(56,99)(57,100)(58,101)(59,102)(60,103)(109,163)(110,164)(111,165)(112,166)(113,167)(114,168)(115,157)(116,158)(117,159)(118,160)(119,161)(120,162)(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)(127,169)(128,170)(129,171)(130,172)(131,173)(132,174)(133,187)(134,188)(135,189)(136,190)(137,191)(138,192)(139,181)(140,182)(141,183)(142,184)(143,185)(144,186), (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), (1,102,15,68)(2,97,16,63)(3,104,17,70)(4,99,18,65)(5,106,19,72)(6,101,20,67)(7,108,21,62)(8,103,22,69)(9,98,23,64)(10,105,24,71)(11,100,13,66)(12,107,14,61)(25,140,166,131)(26,135,167,126)(27,142,168,121)(28,137,157,128)(29,144,158,123)(30,139,159,130)(31,134,160,125)(32,141,161,132)(33,136,162,127)(34,143,163,122)(35,138,164,129)(36,133,165,124)(37,85,59,76)(38,92,60,83)(39,87,49,78)(40,94,50,73)(41,89,51,80)(42,96,52,75)(43,91,53,82)(44,86,54,77)(45,93,55,84)(46,88,56,79)(47,95,57,74)(48,90,58,81)(109,176,154,185)(110,171,155,192)(111,178,156,187)(112,173,145,182)(113,180,146,189)(114,175,147,184)(115,170,148,191)(116,177,149,186)(117,172,150,181)(118,179,151,188)(119,174,152,183)(120,169,153,190), (1,128,15,137)(2,121,16,142)(3,126,17,135)(4,131,18,140)(5,124,19,133)(6,129,20,138)(7,122,21,143)(8,127,22,136)(9,132,23,141)(10,125,24,134)(11,130,13,139)(12,123,14,144)(25,105,166,71)(26,98,167,64)(27,103,168,69)(28,108,157,62)(29,101,158,67)(30,106,159,72)(31,99,160,65)(32,104,161,70)(33,97,162,63)(34,102,163,68)(35,107,164,61)(36,100,165,66)(37,154,59,109)(38,147,60,114)(39,152,49,119)(40,145,50,112)(41,150,51,117)(42,155,52,110)(43,148,53,115)(44,153,54,120)(45,146,55,113)(46,151,56,118)(47,156,57,111)(48,149,58,116)(73,188,94,179)(74,181,95,172)(75,186,96,177)(76,191,85,170)(77,184,86,175)(78,189,87,180)(79,182,88,173)(80,187,89,178)(81,192,90,171)(82,185,91,176)(83,190,92,169)(84,183,93,174) );
G=PermutationGroup([(1,85),(2,86),(3,87),(4,88),(5,89),(6,90),(7,91),(8,92),(9,93),(10,94),(11,95),(12,96),(13,74),(14,75),(15,76),(16,77),(17,78),(18,79),(19,80),(20,81),(21,82),(22,83),(23,84),(24,73),(25,145),(26,146),(27,147),(28,148),(29,149),(30,150),(31,151),(32,152),(33,153),(34,154),(35,155),(36,156),(37,68),(38,69),(39,70),(40,71),(41,72),(42,61),(43,62),(44,63),(45,64),(46,65),(47,66),(48,67),(49,104),(50,105),(51,106),(52,107),(53,108),(54,97),(55,98),(56,99),(57,100),(58,101),(59,102),(60,103),(109,163),(110,164),(111,165),(112,166),(113,167),(114,168),(115,157),(116,158),(117,159),(118,160),(119,161),(120,162),(121,175),(122,176),(123,177),(124,178),(125,179),(126,180),(127,169),(128,170),(129,171),(130,172),(131,173),(132,174),(133,187),(134,188),(135,189),(136,190),(137,191),(138,192),(139,181),(140,182),(141,183),(142,184),(143,185),(144,186)], [(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)], [(1,102,15,68),(2,97,16,63),(3,104,17,70),(4,99,18,65),(5,106,19,72),(6,101,20,67),(7,108,21,62),(8,103,22,69),(9,98,23,64),(10,105,24,71),(11,100,13,66),(12,107,14,61),(25,140,166,131),(26,135,167,126),(27,142,168,121),(28,137,157,128),(29,144,158,123),(30,139,159,130),(31,134,160,125),(32,141,161,132),(33,136,162,127),(34,143,163,122),(35,138,164,129),(36,133,165,124),(37,85,59,76),(38,92,60,83),(39,87,49,78),(40,94,50,73),(41,89,51,80),(42,96,52,75),(43,91,53,82),(44,86,54,77),(45,93,55,84),(46,88,56,79),(47,95,57,74),(48,90,58,81),(109,176,154,185),(110,171,155,192),(111,178,156,187),(112,173,145,182),(113,180,146,189),(114,175,147,184),(115,170,148,191),(116,177,149,186),(117,172,150,181),(118,179,151,188),(119,174,152,183),(120,169,153,190)], [(1,128,15,137),(2,121,16,142),(3,126,17,135),(4,131,18,140),(5,124,19,133),(6,129,20,138),(7,122,21,143),(8,127,22,136),(9,132,23,141),(10,125,24,134),(11,130,13,139),(12,123,14,144),(25,105,166,71),(26,98,167,64),(27,103,168,69),(28,108,157,62),(29,101,158,67),(30,106,159,72),(31,99,160,65),(32,104,161,70),(33,97,162,63),(34,102,163,68),(35,107,164,61),(36,100,165,66),(37,154,59,109),(38,147,60,114),(39,152,49,119),(40,145,50,112),(41,150,51,117),(42,155,52,110),(43,148,53,115),(44,153,54,120),(45,146,55,113),(46,151,56,118),(47,156,57,111),(48,149,58,116),(73,188,94,179),(74,181,95,172),(75,186,96,177),(76,191,85,170),(77,184,86,175),(78,189,87,180),(79,182,88,173),(80,187,89,178),(81,192,90,171),(82,185,91,176),(83,190,92,169),(84,183,93,174)])
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 4Q | 4R | 4S | 4T | 6A | ··· | 6G | 12A | ··· | 12L |
| order | 1 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | - | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | C4○D4 | Dic6 | D4⋊2S3 | Q8⋊3S3 |
| kernel | C2×C12.3Q8 | C12.3Q8 | C2×C4×Dic3 | C2×Dic3⋊C4 | C2×C4⋊Dic3 | C6×C4⋊C4 | C2×C4⋊C4 | C2×C12 | C4⋊C4 | C22×C4 | C2×C6 | C2×C4 | C22 | C22 |
| # reps | 1 | 8 | 1 | 2 | 3 | 1 | 1 | 4 | 4 | 3 | 8 | 8 | 2 | 2 |
Matrix representation of C2×C12.3Q8 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 3 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 12 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 4 |
| 0 | 0 | 0 | 0 | 7 | 7 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 5 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 11 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 12 | 0 | 0 |
| 0 | 0 | 3 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 6 |
| 0 | 0 | 0 | 0 | 4 | 4 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,3,0,0,0,0,0,10,0,0,0,0,0,0,3,12,0,0,0,0,0,9,0,0,0,0,0,0,6,7,0,0,0,0,4,7],[8,5,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,11,1],[1,12,0,0,0,0,2,12,0,0,0,0,0,0,11,3,0,0,0,0,12,2,0,0,0,0,0,0,9,4,0,0,0,0,6,4] >;
C2×C12.3Q8 in GAP, Magma, Sage, TeX
C_2\times C_{12}._3Q_8 % in TeX
G:=Group("C2xC12.3Q8"); // GroupNames label
G:=SmallGroup(192,1058);
// by ID
G=gap.SmallGroup(192,1058);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,184,1571,297,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^7,d*b*d^-1=b^5,d*c*d^-1=b^6*c^-1>;
// generators/relations