metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊7Dic6, C42.124D6, C6.652- (1+4), (C3×Q8)⋊7Q8, C4⋊C4.293D6, C3⋊3(Q8⋊3Q8), (C4×Q8).19S3, C12.46(C2×Q8), (Q8×C12).13C2, (C2×Q8).222D6, C4.19(C2×Dic6), C6.17(C22×Q8), (C2×C6).114C24, C2.22(Q8○D12), C12⋊2Q8.25C2, (Q8×Dic3).12C2, (C4×Dic6).23C2, C12.333(C4○D4), (C2×C12).168C23, (C4×C12).166C22, C4.49(Q8⋊3S3), C4⋊Dic3.43C22, C12.3Q8.11C2, (C6×Q8).214C22, C2.19(C22×Dic6), C22.139(S3×C23), (C2×Dic3).52C23, (C4×Dic3).83C22, Dic3⋊C4.115C22, (C2×Dic6).242C22, C6.110(C2×C4○D4), C2.10(C2×Q8⋊3S3), (C3×C4⋊C4).342C22, (C2×C4).733(C22×S3), SmallGroup(192,1129)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 392 in 200 conjugacy classes, 115 normal (18 characteristic)
C1, C2 [×3], C3, C4 [×8], C4 [×11], C22, C6 [×3], C2×C4, C2×C4 [×6], C2×C4 [×8], Q8 [×4], Q8 [×6], Dic3 [×8], C12 [×8], C12 [×3], C2×C6, C42 [×3], C42 [×6], C4⋊C4 [×3], C4⋊C4 [×19], C2×Q8, C2×Q8 [×3], Dic6 [×6], C2×Dic3 [×8], C2×C12, C2×C12 [×6], C3×Q8 [×4], C4×Q8, C4×Q8 [×5], C42.C2 [×6], C4⋊Q8 [×3], C4×Dic3 [×6], Dic3⋊C4 [×6], C4⋊Dic3, C4⋊Dic3 [×12], C4×C12 [×3], C3×C4⋊C4 [×3], C2×Dic6 [×3], C6×Q8, Q8⋊3Q8, C4×Dic6 [×3], C12⋊2Q8 [×3], C12.3Q8 [×6], Q8×Dic3 [×2], Q8×C12, Q8⋊7Dic6
Quotients:
C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D6 [×7], C2×Q8 [×6], C4○D4 [×2], C24, Dic6 [×4], C22×S3 [×7], C22×Q8, C2×C4○D4, 2- (1+4), C2×Dic6 [×6], Q8⋊3S3 [×2], S3×C23, Q8⋊3Q8, C22×Dic6, C2×Q8⋊3S3, Q8○D12, Q8⋊7Dic6
Generators and relations
G = < a,b,c,d | a4=c12=1, b2=a2, d2=c6, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=a2b, dcd-1=c-1 >
►Regular action on 192 points
Generators in S192
(1 130 192 34)(2 131 181 35)(3 132 182 36)(4 121 183 25)(5 122 184 26)(6 123 185 27)(7 124 186 28)(8 125 187 29)(9 126 188 30)(10 127 189 31)(11 128 190 32)(12 129 191 33)(13 80 102 53)(14 81 103 54)(15 82 104 55)(16 83 105 56)(17 84 106 57)(18 73 107 58)(19 74 108 59)(20 75 97 60)(21 76 98 49)(22 77 99 50)(23 78 100 51)(24 79 101 52)(37 71 135 148)(38 72 136 149)(39 61 137 150)(40 62 138 151)(41 63 139 152)(42 64 140 153)(43 65 141 154)(44 66 142 155)(45 67 143 156)(46 68 144 145)(47 69 133 146)(48 70 134 147)(85 110 169 157)(86 111 170 158)(87 112 171 159)(88 113 172 160)(89 114 173 161)(90 115 174 162)(91 116 175 163)(92 117 176 164)(93 118 177 165)(94 119 178 166)(95 120 179 167)(96 109 180 168)
(1 96 192 180)(2 85 181 169)(3 86 182 170)(4 87 183 171)(5 88 184 172)(6 89 185 173)(7 90 186 174)(8 91 187 175)(9 92 188 176)(10 93 189 177)(11 94 190 178)(12 95 191 179)(13 152 102 63)(14 153 103 64)(15 154 104 65)(16 155 105 66)(17 156 106 67)(18 145 107 68)(19 146 108 69)(20 147 97 70)(21 148 98 71)(22 149 99 72)(23 150 100 61)(24 151 101 62)(25 112 121 159)(26 113 122 160)(27 114 123 161)(28 115 124 162)(29 116 125 163)(30 117 126 164)(31 118 127 165)(32 119 128 166)(33 120 129 167)(34 109 130 168)(35 110 131 157)(36 111 132 158)(37 76 135 49)(38 77 136 50)(39 78 137 51)(40 79 138 52)(41 80 139 53)(42 81 140 54)(43 82 141 55)(44 83 142 56)(45 84 143 57)(46 73 144 58)(47 74 133 59)(48 75 134 60)
(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 16 7 22)(2 15 8 21)(3 14 9 20)(4 13 10 19)(5 24 11 18)(6 23 12 17)(25 53 31 59)(26 52 32 58)(27 51 33 57)(28 50 34 56)(29 49 35 55)(30 60 36 54)(37 157 43 163)(38 168 44 162)(39 167 45 161)(40 166 46 160)(41 165 47 159)(42 164 48 158)(61 95 67 89)(62 94 68 88)(63 93 69 87)(64 92 70 86)(65 91 71 85)(66 90 72 96)(73 122 79 128)(74 121 80 127)(75 132 81 126)(76 131 82 125)(77 130 83 124)(78 129 84 123)(97 182 103 188)(98 181 104 187)(99 192 105 186)(100 191 106 185)(101 190 107 184)(102 189 108 183)(109 142 115 136)(110 141 116 135)(111 140 117 134)(112 139 118 133)(113 138 119 144)(114 137 120 143)(145 172 151 178)(146 171 152 177)(147 170 153 176)(148 169 154 175)(149 180 155 174)(150 179 156 173)
G:=sub<Sym(192)| (1,130,192,34)(2,131,181,35)(3,132,182,36)(4,121,183,25)(5,122,184,26)(6,123,185,27)(7,124,186,28)(8,125,187,29)(9,126,188,30)(10,127,189,31)(11,128,190,32)(12,129,191,33)(13,80,102,53)(14,81,103,54)(15,82,104,55)(16,83,105,56)(17,84,106,57)(18,73,107,58)(19,74,108,59)(20,75,97,60)(21,76,98,49)(22,77,99,50)(23,78,100,51)(24,79,101,52)(37,71,135,148)(38,72,136,149)(39,61,137,150)(40,62,138,151)(41,63,139,152)(42,64,140,153)(43,65,141,154)(44,66,142,155)(45,67,143,156)(46,68,144,145)(47,69,133,146)(48,70,134,147)(85,110,169,157)(86,111,170,158)(87,112,171,159)(88,113,172,160)(89,114,173,161)(90,115,174,162)(91,116,175,163)(92,117,176,164)(93,118,177,165)(94,119,178,166)(95,120,179,167)(96,109,180,168), (1,96,192,180)(2,85,181,169)(3,86,182,170)(4,87,183,171)(5,88,184,172)(6,89,185,173)(7,90,186,174)(8,91,187,175)(9,92,188,176)(10,93,189,177)(11,94,190,178)(12,95,191,179)(13,152,102,63)(14,153,103,64)(15,154,104,65)(16,155,105,66)(17,156,106,67)(18,145,107,68)(19,146,108,69)(20,147,97,70)(21,148,98,71)(22,149,99,72)(23,150,100,61)(24,151,101,62)(25,112,121,159)(26,113,122,160)(27,114,123,161)(28,115,124,162)(29,116,125,163)(30,117,126,164)(31,118,127,165)(32,119,128,166)(33,120,129,167)(34,109,130,168)(35,110,131,157)(36,111,132,158)(37,76,135,49)(38,77,136,50)(39,78,137,51)(40,79,138,52)(41,80,139,53)(42,81,140,54)(43,82,141,55)(44,83,142,56)(45,84,143,57)(46,73,144,58)(47,74,133,59)(48,75,134,60), (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), (1,16,7,22)(2,15,8,21)(3,14,9,20)(4,13,10,19)(5,24,11,18)(6,23,12,17)(25,53,31,59)(26,52,32,58)(27,51,33,57)(28,50,34,56)(29,49,35,55)(30,60,36,54)(37,157,43,163)(38,168,44,162)(39,167,45,161)(40,166,46,160)(41,165,47,159)(42,164,48,158)(61,95,67,89)(62,94,68,88)(63,93,69,87)(64,92,70,86)(65,91,71,85)(66,90,72,96)(73,122,79,128)(74,121,80,127)(75,132,81,126)(76,131,82,125)(77,130,83,124)(78,129,84,123)(97,182,103,188)(98,181,104,187)(99,192,105,186)(100,191,106,185)(101,190,107,184)(102,189,108,183)(109,142,115,136)(110,141,116,135)(111,140,117,134)(112,139,118,133)(113,138,119,144)(114,137,120,143)(145,172,151,178)(146,171,152,177)(147,170,153,176)(148,169,154,175)(149,180,155,174)(150,179,156,173)>;
G:=Group( (1,130,192,34)(2,131,181,35)(3,132,182,36)(4,121,183,25)(5,122,184,26)(6,123,185,27)(7,124,186,28)(8,125,187,29)(9,126,188,30)(10,127,189,31)(11,128,190,32)(12,129,191,33)(13,80,102,53)(14,81,103,54)(15,82,104,55)(16,83,105,56)(17,84,106,57)(18,73,107,58)(19,74,108,59)(20,75,97,60)(21,76,98,49)(22,77,99,50)(23,78,100,51)(24,79,101,52)(37,71,135,148)(38,72,136,149)(39,61,137,150)(40,62,138,151)(41,63,139,152)(42,64,140,153)(43,65,141,154)(44,66,142,155)(45,67,143,156)(46,68,144,145)(47,69,133,146)(48,70,134,147)(85,110,169,157)(86,111,170,158)(87,112,171,159)(88,113,172,160)(89,114,173,161)(90,115,174,162)(91,116,175,163)(92,117,176,164)(93,118,177,165)(94,119,178,166)(95,120,179,167)(96,109,180,168), (1,96,192,180)(2,85,181,169)(3,86,182,170)(4,87,183,171)(5,88,184,172)(6,89,185,173)(7,90,186,174)(8,91,187,175)(9,92,188,176)(10,93,189,177)(11,94,190,178)(12,95,191,179)(13,152,102,63)(14,153,103,64)(15,154,104,65)(16,155,105,66)(17,156,106,67)(18,145,107,68)(19,146,108,69)(20,147,97,70)(21,148,98,71)(22,149,99,72)(23,150,100,61)(24,151,101,62)(25,112,121,159)(26,113,122,160)(27,114,123,161)(28,115,124,162)(29,116,125,163)(30,117,126,164)(31,118,127,165)(32,119,128,166)(33,120,129,167)(34,109,130,168)(35,110,131,157)(36,111,132,158)(37,76,135,49)(38,77,136,50)(39,78,137,51)(40,79,138,52)(41,80,139,53)(42,81,140,54)(43,82,141,55)(44,83,142,56)(45,84,143,57)(46,73,144,58)(47,74,133,59)(48,75,134,60), (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), (1,16,7,22)(2,15,8,21)(3,14,9,20)(4,13,10,19)(5,24,11,18)(6,23,12,17)(25,53,31,59)(26,52,32,58)(27,51,33,57)(28,50,34,56)(29,49,35,55)(30,60,36,54)(37,157,43,163)(38,168,44,162)(39,167,45,161)(40,166,46,160)(41,165,47,159)(42,164,48,158)(61,95,67,89)(62,94,68,88)(63,93,69,87)(64,92,70,86)(65,91,71,85)(66,90,72,96)(73,122,79,128)(74,121,80,127)(75,132,81,126)(76,131,82,125)(77,130,83,124)(78,129,84,123)(97,182,103,188)(98,181,104,187)(99,192,105,186)(100,191,106,185)(101,190,107,184)(102,189,108,183)(109,142,115,136)(110,141,116,135)(111,140,117,134)(112,139,118,133)(113,138,119,144)(114,137,120,143)(145,172,151,178)(146,171,152,177)(147,170,153,176)(148,169,154,175)(149,180,155,174)(150,179,156,173) );
G=PermutationGroup([(1,130,192,34),(2,131,181,35),(3,132,182,36),(4,121,183,25),(5,122,184,26),(6,123,185,27),(7,124,186,28),(8,125,187,29),(9,126,188,30),(10,127,189,31),(11,128,190,32),(12,129,191,33),(13,80,102,53),(14,81,103,54),(15,82,104,55),(16,83,105,56),(17,84,106,57),(18,73,107,58),(19,74,108,59),(20,75,97,60),(21,76,98,49),(22,77,99,50),(23,78,100,51),(24,79,101,52),(37,71,135,148),(38,72,136,149),(39,61,137,150),(40,62,138,151),(41,63,139,152),(42,64,140,153),(43,65,141,154),(44,66,142,155),(45,67,143,156),(46,68,144,145),(47,69,133,146),(48,70,134,147),(85,110,169,157),(86,111,170,158),(87,112,171,159),(88,113,172,160),(89,114,173,161),(90,115,174,162),(91,116,175,163),(92,117,176,164),(93,118,177,165),(94,119,178,166),(95,120,179,167),(96,109,180,168)], [(1,96,192,180),(2,85,181,169),(3,86,182,170),(4,87,183,171),(5,88,184,172),(6,89,185,173),(7,90,186,174),(8,91,187,175),(9,92,188,176),(10,93,189,177),(11,94,190,178),(12,95,191,179),(13,152,102,63),(14,153,103,64),(15,154,104,65),(16,155,105,66),(17,156,106,67),(18,145,107,68),(19,146,108,69),(20,147,97,70),(21,148,98,71),(22,149,99,72),(23,150,100,61),(24,151,101,62),(25,112,121,159),(26,113,122,160),(27,114,123,161),(28,115,124,162),(29,116,125,163),(30,117,126,164),(31,118,127,165),(32,119,128,166),(33,120,129,167),(34,109,130,168),(35,110,131,157),(36,111,132,158),(37,76,135,49),(38,77,136,50),(39,78,137,51),(40,79,138,52),(41,80,139,53),(42,81,140,54),(43,82,141,55),(44,83,142,56),(45,84,143,57),(46,73,144,58),(47,74,133,59),(48,75,134,60)], [(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)], [(1,16,7,22),(2,15,8,21),(3,14,9,20),(4,13,10,19),(5,24,11,18),(6,23,12,17),(25,53,31,59),(26,52,32,58),(27,51,33,57),(28,50,34,56),(29,49,35,55),(30,60,36,54),(37,157,43,163),(38,168,44,162),(39,167,45,161),(40,166,46,160),(41,165,47,159),(42,164,48,158),(61,95,67,89),(62,94,68,88),(63,93,69,87),(64,92,70,86),(65,91,71,85),(66,90,72,96),(73,122,79,128),(74,121,80,127),(75,132,81,126),(76,131,82,125),(77,130,83,124),(78,129,84,123),(97,182,103,188),(98,181,104,187),(99,192,105,186),(100,191,106,185),(101,190,107,184),(102,189,108,183),(109,142,115,136),(110,141,116,135),(111,140,117,134),(112,139,118,133),(113,138,119,144),(114,137,120,143),(145,172,151,178),(146,171,152,177),(147,170,153,176),(148,169,154,175),(149,180,155,174),(150,179,156,173)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 12 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 6 |
| 0 | 0 | 4 | 1 |
| 6 | 3 | 0 | 0 |
| 10 | 3 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 11 | 2 | 0 | 0 |
| 4 | 2 | 0 | 0 |
| 0 | 0 | 5 | 10 |
| 0 | 0 | 8 | 8 |
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,1,12,0,0,2,12],[12,0,0,0,0,12,0,0,0,0,12,4,0,0,6,1],[6,10,0,0,3,3,0,0,0,0,12,0,0,0,0,12],[11,4,0,0,2,2,0,0,0,0,5,8,0,0,10,8] >;
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | ··· | 4U | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | D6 | C4○D4 | Dic6 | 2- (1+4) | Q8⋊3S3 | Q8○D12 |
| kernel | Q8⋊7Dic6 | C4×Dic6 | C12⋊2Q8 | C12.3Q8 | Q8×Dic3 | Q8×C12 | C4×Q8 | C3×Q8 | C42 | C4⋊C4 | C2×Q8 | C12 | Q8 | C6 | C4 | C2 |
| # reps | 1 | 3 | 3 | 6 | 2 | 1 | 1 | 4 | 3 | 3 | 1 | 4 | 8 | 1 | 2 | 2 |
In GAP, Magma, Sage, TeX
Q_8\rtimes_7Dic_6
% in TeX
G:=Group("Q8:7Dic6"); // GroupNames label
G:=SmallGroup(192,1129);
// by ID
G=gap.SmallGroup(192,1129);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,232,387,184,1571,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^12=1,b^2=a^2,d^2=c^6,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀