direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C3⋊Q32, C6⋊2Q32, C12.25D8, C24.25D4, Q16.8D6, C24.28C23, Dic12.12C22, C3⋊3(C2×Q32), C6.67(C2×D8), (C2×C6).46D8, (C2×C8).240D6, C3⋊C16.9C22, (C2×Q16).2S3, (C6×Q16).3C2, C4.11(D4⋊S3), (C2×C12).184D4, C12.183(C2×D4), C8.17(C3⋊D4), C8.34(C22×S3), (C2×C24).92C22, (C3×Q16).8C22, C22.24(D4⋊S3), (C2×Dic12).10C2, (C2×C3⋊C16).6C2, C2.22(C2×D4⋊S3), C4.13(C2×C3⋊D4), (C2×C4).145(C3⋊D4), SmallGroup(192,739)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C3⋊Q32
G = < a,b,c,d | a2=b3=c16=1, d2=c8, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >
Subgroups: 216 in 82 conjugacy classes, 39 normal (23 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, Q8, Dic3, C12, C12, C2×C6, C16, C2×C8, Q16, Q16, C2×Q8, C24, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C2×C16, Q32, C2×Q16, C2×Q16, C3⋊C16, Dic12, Dic12, C2×C24, C3×Q16, C3×Q16, C2×Dic6, C6×Q8, C2×Q32, C2×C3⋊C16, C3⋊Q32, C2×Dic12, C6×Q16, C2×C3⋊Q32
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C3⋊D4, C22×S3, Q32, C2×D8, D4⋊S3, C2×C3⋊D4, C2×Q32, C3⋊Q32, C2×D4⋊S3, C2×C3⋊Q32
►Regular action on 192 points
Generators in S192
(1 75)(2 76)(3 77)(4 78)(5 79)(6 80)(7 65)(8 66)(9 67)(10 68)(11 69)(12 70)(13 71)(14 72)(15 73)(16 74)(17 119)(18 120)(19 121)(20 122)(21 123)(22 124)(23 125)(24 126)(25 127)(26 128)(27 113)(28 114)(29 115)(30 116)(31 117)(32 118)(33 183)(34 184)(35 185)(36 186)(37 187)(38 188)(39 189)(40 190)(41 191)(42 192)(43 177)(44 178)(45 179)(46 180)(47 181)(48 182)(49 170)(50 171)(51 172)(52 173)(53 174)(54 175)(55 176)(56 161)(57 162)(58 163)(59 164)(60 165)(61 166)(62 167)(63 168)(64 169)(81 159)(82 160)(83 145)(84 146)(85 147)(86 148)(87 149)(88 150)(89 151)(90 152)(91 153)(92 154)(93 155)(94 156)(95 157)(96 158)(97 135)(98 136)(99 137)(100 138)(101 139)(102 140)(103 141)(104 142)(105 143)(106 144)(107 129)(108 130)(109 131)(110 132)(111 133)(112 134)
(1 89 173)(2 174 90)(3 91 175)(4 176 92)(5 93 161)(6 162 94)(7 95 163)(8 164 96)(9 81 165)(10 166 82)(11 83 167)(12 168 84)(13 85 169)(14 170 86)(15 87 171)(16 172 88)(17 138 45)(18 46 139)(19 140 47)(20 48 141)(21 142 33)(22 34 143)(23 144 35)(24 36 129)(25 130 37)(26 38 131)(27 132 39)(28 40 133)(29 134 41)(30 42 135)(31 136 43)(32 44 137)(49 148 72)(50 73 149)(51 150 74)(52 75 151)(53 152 76)(54 77 153)(55 154 78)(56 79 155)(57 156 80)(58 65 157)(59 158 66)(60 67 159)(61 160 68)(62 69 145)(63 146 70)(64 71 147)(97 116 192)(98 177 117)(99 118 178)(100 179 119)(101 120 180)(102 181 121)(103 122 182)(104 183 123)(105 124 184)(106 185 125)(107 126 186)(108 187 127)(109 128 188)(110 189 113)(111 114 190)(112 191 115)
(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 40 9 48)(2 39 10 47)(3 38 11 46)(4 37 12 45)(5 36 13 44)(6 35 14 43)(7 34 15 42)(8 33 16 41)(17 176 25 168)(18 175 26 167)(19 174 27 166)(20 173 28 165)(21 172 29 164)(22 171 30 163)(23 170 31 162)(24 169 32 161)(49 117 57 125)(50 116 58 124)(51 115 59 123)(52 114 60 122)(53 113 61 121)(54 128 62 120)(55 127 63 119)(56 126 64 118)(65 184 73 192)(66 183 74 191)(67 182 75 190)(68 181 76 189)(69 180 77 188)(70 179 78 187)(71 178 79 186)(72 177 80 185)(81 141 89 133)(82 140 90 132)(83 139 91 131)(84 138 92 130)(85 137 93 129)(86 136 94 144)(87 135 95 143)(88 134 96 142)(97 157 105 149)(98 156 106 148)(99 155 107 147)(100 154 108 146)(101 153 109 145)(102 152 110 160)(103 151 111 159)(104 150 112 158)
G:=sub<Sym(192)| (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,73)(16,74)(17,119)(18,120)(19,121)(20,122)(21,123)(22,124)(23,125)(24,126)(25,127)(26,128)(27,113)(28,114)(29,115)(30,116)(31,117)(32,118)(33,183)(34,184)(35,185)(36,186)(37,187)(38,188)(39,189)(40,190)(41,191)(42,192)(43,177)(44,178)(45,179)(46,180)(47,181)(48,182)(49,170)(50,171)(51,172)(52,173)(53,174)(54,175)(55,176)(56,161)(57,162)(58,163)(59,164)(60,165)(61,166)(62,167)(63,168)(64,169)(81,159)(82,160)(83,145)(84,146)(85,147)(86,148)(87,149)(88,150)(89,151)(90,152)(91,153)(92,154)(93,155)(94,156)(95,157)(96,158)(97,135)(98,136)(99,137)(100,138)(101,139)(102,140)(103,141)(104,142)(105,143)(106,144)(107,129)(108,130)(109,131)(110,132)(111,133)(112,134), (1,89,173)(2,174,90)(3,91,175)(4,176,92)(5,93,161)(6,162,94)(7,95,163)(8,164,96)(9,81,165)(10,166,82)(11,83,167)(12,168,84)(13,85,169)(14,170,86)(15,87,171)(16,172,88)(17,138,45)(18,46,139)(19,140,47)(20,48,141)(21,142,33)(22,34,143)(23,144,35)(24,36,129)(25,130,37)(26,38,131)(27,132,39)(28,40,133)(29,134,41)(30,42,135)(31,136,43)(32,44,137)(49,148,72)(50,73,149)(51,150,74)(52,75,151)(53,152,76)(54,77,153)(55,154,78)(56,79,155)(57,156,80)(58,65,157)(59,158,66)(60,67,159)(61,160,68)(62,69,145)(63,146,70)(64,71,147)(97,116,192)(98,177,117)(99,118,178)(100,179,119)(101,120,180)(102,181,121)(103,122,182)(104,183,123)(105,124,184)(106,185,125)(107,126,186)(108,187,127)(109,128,188)(110,189,113)(111,114,190)(112,191,115), (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), (1,40,9,48)(2,39,10,47)(3,38,11,46)(4,37,12,45)(5,36,13,44)(6,35,14,43)(7,34,15,42)(8,33,16,41)(17,176,25,168)(18,175,26,167)(19,174,27,166)(20,173,28,165)(21,172,29,164)(22,171,30,163)(23,170,31,162)(24,169,32,161)(49,117,57,125)(50,116,58,124)(51,115,59,123)(52,114,60,122)(53,113,61,121)(54,128,62,120)(55,127,63,119)(56,126,64,118)(65,184,73,192)(66,183,74,191)(67,182,75,190)(68,181,76,189)(69,180,77,188)(70,179,78,187)(71,178,79,186)(72,177,80,185)(81,141,89,133)(82,140,90,132)(83,139,91,131)(84,138,92,130)(85,137,93,129)(86,136,94,144)(87,135,95,143)(88,134,96,142)(97,157,105,149)(98,156,106,148)(99,155,107,147)(100,154,108,146)(101,153,109,145)(102,152,110,160)(103,151,111,159)(104,150,112,158)>;
G:=Group( (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,73)(16,74)(17,119)(18,120)(19,121)(20,122)(21,123)(22,124)(23,125)(24,126)(25,127)(26,128)(27,113)(28,114)(29,115)(30,116)(31,117)(32,118)(33,183)(34,184)(35,185)(36,186)(37,187)(38,188)(39,189)(40,190)(41,191)(42,192)(43,177)(44,178)(45,179)(46,180)(47,181)(48,182)(49,170)(50,171)(51,172)(52,173)(53,174)(54,175)(55,176)(56,161)(57,162)(58,163)(59,164)(60,165)(61,166)(62,167)(63,168)(64,169)(81,159)(82,160)(83,145)(84,146)(85,147)(86,148)(87,149)(88,150)(89,151)(90,152)(91,153)(92,154)(93,155)(94,156)(95,157)(96,158)(97,135)(98,136)(99,137)(100,138)(101,139)(102,140)(103,141)(104,142)(105,143)(106,144)(107,129)(108,130)(109,131)(110,132)(111,133)(112,134), (1,89,173)(2,174,90)(3,91,175)(4,176,92)(5,93,161)(6,162,94)(7,95,163)(8,164,96)(9,81,165)(10,166,82)(11,83,167)(12,168,84)(13,85,169)(14,170,86)(15,87,171)(16,172,88)(17,138,45)(18,46,139)(19,140,47)(20,48,141)(21,142,33)(22,34,143)(23,144,35)(24,36,129)(25,130,37)(26,38,131)(27,132,39)(28,40,133)(29,134,41)(30,42,135)(31,136,43)(32,44,137)(49,148,72)(50,73,149)(51,150,74)(52,75,151)(53,152,76)(54,77,153)(55,154,78)(56,79,155)(57,156,80)(58,65,157)(59,158,66)(60,67,159)(61,160,68)(62,69,145)(63,146,70)(64,71,147)(97,116,192)(98,177,117)(99,118,178)(100,179,119)(101,120,180)(102,181,121)(103,122,182)(104,183,123)(105,124,184)(106,185,125)(107,126,186)(108,187,127)(109,128,188)(110,189,113)(111,114,190)(112,191,115), (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), (1,40,9,48)(2,39,10,47)(3,38,11,46)(4,37,12,45)(5,36,13,44)(6,35,14,43)(7,34,15,42)(8,33,16,41)(17,176,25,168)(18,175,26,167)(19,174,27,166)(20,173,28,165)(21,172,29,164)(22,171,30,163)(23,170,31,162)(24,169,32,161)(49,117,57,125)(50,116,58,124)(51,115,59,123)(52,114,60,122)(53,113,61,121)(54,128,62,120)(55,127,63,119)(56,126,64,118)(65,184,73,192)(66,183,74,191)(67,182,75,190)(68,181,76,189)(69,180,77,188)(70,179,78,187)(71,178,79,186)(72,177,80,185)(81,141,89,133)(82,140,90,132)(83,139,91,131)(84,138,92,130)(85,137,93,129)(86,136,94,144)(87,135,95,143)(88,134,96,142)(97,157,105,149)(98,156,106,148)(99,155,107,147)(100,154,108,146)(101,153,109,145)(102,152,110,160)(103,151,111,159)(104,150,112,158) );
G=PermutationGroup([[(1,75),(2,76),(3,77),(4,78),(5,79),(6,80),(7,65),(8,66),(9,67),(10,68),(11,69),(12,70),(13,71),(14,72),(15,73),(16,74),(17,119),(18,120),(19,121),(20,122),(21,123),(22,124),(23,125),(24,126),(25,127),(26,128),(27,113),(28,114),(29,115),(30,116),(31,117),(32,118),(33,183),(34,184),(35,185),(36,186),(37,187),(38,188),(39,189),(40,190),(41,191),(42,192),(43,177),(44,178),(45,179),(46,180),(47,181),(48,182),(49,170),(50,171),(51,172),(52,173),(53,174),(54,175),(55,176),(56,161),(57,162),(58,163),(59,164),(60,165),(61,166),(62,167),(63,168),(64,169),(81,159),(82,160),(83,145),(84,146),(85,147),(86,148),(87,149),(88,150),(89,151),(90,152),(91,153),(92,154),(93,155),(94,156),(95,157),(96,158),(97,135),(98,136),(99,137),(100,138),(101,139),(102,140),(103,141),(104,142),(105,143),(106,144),(107,129),(108,130),(109,131),(110,132),(111,133),(112,134)], [(1,89,173),(2,174,90),(3,91,175),(4,176,92),(5,93,161),(6,162,94),(7,95,163),(8,164,96),(9,81,165),(10,166,82),(11,83,167),(12,168,84),(13,85,169),(14,170,86),(15,87,171),(16,172,88),(17,138,45),(18,46,139),(19,140,47),(20,48,141),(21,142,33),(22,34,143),(23,144,35),(24,36,129),(25,130,37),(26,38,131),(27,132,39),(28,40,133),(29,134,41),(30,42,135),(31,136,43),(32,44,137),(49,148,72),(50,73,149),(51,150,74),(52,75,151),(53,152,76),(54,77,153),(55,154,78),(56,79,155),(57,156,80),(58,65,157),(59,158,66),(60,67,159),(61,160,68),(62,69,145),(63,146,70),(64,71,147),(97,116,192),(98,177,117),(99,118,178),(100,179,119),(101,120,180),(102,181,121),(103,122,182),(104,183,123),(105,124,184),(106,185,125),(107,126,186),(108,187,127),(109,128,188),(110,189,113),(111,114,190),(112,191,115)], [(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)], [(1,40,9,48),(2,39,10,47),(3,38,11,46),(4,37,12,45),(5,36,13,44),(6,35,14,43),(7,34,15,42),(8,33,16,41),(17,176,25,168),(18,175,26,167),(19,174,27,166),(20,173,28,165),(21,172,29,164),(22,171,30,163),(23,170,31,162),(24,169,32,161),(49,117,57,125),(50,116,58,124),(51,115,59,123),(52,114,60,122),(53,113,61,121),(54,128,62,120),(55,127,63,119),(56,126,64,118),(65,184,73,192),(66,183,74,191),(67,182,75,190),(68,181,76,189),(69,180,77,188),(70,179,78,187),(71,178,79,186),(72,177,80,185),(81,141,89,133),(82,140,90,132),(83,139,91,131),(84,138,92,130),(85,137,93,129),(86,136,94,144),(87,135,95,143),(88,134,96,142),(97,157,105,149),(98,156,106,148),(99,155,107,147),(100,154,108,146),(101,153,109,145),(102,152,110,160),(103,151,111,159),(104,150,112,158)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 16A | ··· | 16H | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 8 | 8 | 24 | 24 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 6 | ··· | 6 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D8 | D8 | C3⋊D4 | C3⋊D4 | Q32 | D4⋊S3 | D4⋊S3 | C3⋊Q32 |
| kernel | C2×C3⋊Q32 | C2×C3⋊C16 | C3⋊Q32 | C2×Dic12 | C6×Q16 | C2×Q16 | C24 | C2×C12 | C2×C8 | Q16 | C12 | C2×C6 | C8 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 8 | 1 | 1 | 4 |
Matrix representation of C2×C3⋊Q32 ►in GL4(𝔽97) generated by
| 96 | 0 | 0 | 0 |
| 0 | 96 | 0 | 0 |
| 0 | 0 | 96 | 0 |
| 0 | 0 | 0 | 96 |
| 96 | 96 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 36 | 89 | 0 | 0 |
| 53 | 61 | 0 | 0 |
| 0 | 0 | 69 | 19 |
| 0 | 0 | 82 | 24 |
| 56 | 15 | 0 | 0 |
| 82 | 41 | 0 | 0 |
| 0 | 0 | 26 | 89 |
| 0 | 0 | 24 | 71 |
G:=sub<GL(4,GF(97))| [96,0,0,0,0,96,0,0,0,0,96,0,0,0,0,96],[96,1,0,0,96,0,0,0,0,0,1,0,0,0,0,1],[36,53,0,0,89,61,0,0,0,0,69,82,0,0,19,24],[56,82,0,0,15,41,0,0,0,0,26,24,0,0,89,71] >;
C2×C3⋊Q32 in GAP, Magma, Sage, TeX
C_2\times C_3\rtimes Q_{32} % in TeX
G:=Group("C2xC3:Q32"); // GroupNames label
G:=SmallGroup(192,739);
// by ID
G=gap.SmallGroup(192,739);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,254,184,675,185,192,1684,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^3=c^16=1,d^2=c^8,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations