direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C3⋊C32, C6⋊C32, C48.4C4, C24.5C8, C12.3C16, C16.21D6, C16.4Dic3, C48.26C22, C3⋊2(C2×C32), C8.6(C3⋊C8), C4.3(C3⋊C16), C6.8(C2×C16), (C2×C6).2C16, (C2×C16).9S3, (C2×C12).11C8, (C2×C48).13C2, C12.42(C2×C8), (C2×C24).25C4, C24.80(C2×C4), C22.2(C3⋊C16), C8.21(C2×Dic3), (C2×C8).18Dic3, C2.2(C2×C3⋊C16), C4.14(C2×C3⋊C8), (C2×C4).8(C3⋊C8), SmallGroup(192,57)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C2×C3⋊C32 |
Generators and relations for C2×C3⋊C32
G = < a,b,c | a2=b3=c32=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×C3⋊C32
►Regular action on 192 points
Generators in S192
(1 148)(2 149)(3 150)(4 151)(5 152)(6 153)(7 154)(8 155)(9 156)(10 157)(11 158)(12 159)(13 160)(14 129)(15 130)(16 131)(17 132)(18 133)(19 134)(20 135)(21 136)(22 137)(23 138)(24 139)(25 140)(26 141)(27 142)(28 143)(29 144)(30 145)(31 146)(32 147)(33 96)(34 65)(35 66)(36 67)(37 68)(38 69)(39 70)(40 71)(41 72)(42 73)(43 74)(44 75)(45 76)(46 77)(47 78)(48 79)(49 80)(50 81)(51 82)(52 83)(53 84)(54 85)(55 86)(56 87)(57 88)(58 89)(59 90)(60 91)(61 92)(62 93)(63 94)(64 95)(97 179)(98 180)(99 181)(100 182)(101 183)(102 184)(103 185)(104 186)(105 187)(106 188)(107 189)(108 190)(109 191)(110 192)(111 161)(112 162)(113 163)(114 164)(115 165)(116 166)(117 167)(118 168)(119 169)(120 170)(121 171)(122 172)(123 173)(124 174)(125 175)(126 176)(127 177)(128 178)
(1 44 109)(2 110 45)(3 46 111)(4 112 47)(5 48 113)(6 114 49)(7 50 115)(8 116 51)(9 52 117)(10 118 53)(11 54 119)(12 120 55)(13 56 121)(14 122 57)(15 58 123)(16 124 59)(17 60 125)(18 126 61)(19 62 127)(20 128 63)(21 64 97)(22 98 33)(23 34 99)(24 100 35)(25 36 101)(26 102 37)(27 38 103)(28 104 39)(29 40 105)(30 106 41)(31 42 107)(32 108 43)(65 181 138)(66 139 182)(67 183 140)(68 141 184)(69 185 142)(70 143 186)(71 187 144)(72 145 188)(73 189 146)(74 147 190)(75 191 148)(76 149 192)(77 161 150)(78 151 162)(79 163 152)(80 153 164)(81 165 154)(82 155 166)(83 167 156)(84 157 168)(85 169 158)(86 159 170)(87 171 160)(88 129 172)(89 173 130)(90 131 174)(91 175 132)(92 133 176)(93 177 134)(94 135 178)(95 179 136)(96 137 180)
(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)
G:=sub<Sym(192)| (1,148)(2,149)(3,150)(4,151)(5,152)(6,153)(7,154)(8,155)(9,156)(10,157)(11,158)(12,159)(13,160)(14,129)(15,130)(16,131)(17,132)(18,133)(19,134)(20,135)(21,136)(22,137)(23,138)(24,139)(25,140)(26,141)(27,142)(28,143)(29,144)(30,145)(31,146)(32,147)(33,96)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,71)(41,72)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79)(49,80)(50,81)(51,82)(52,83)(53,84)(54,85)(55,86)(56,87)(57,88)(58,89)(59,90)(60,91)(61,92)(62,93)(63,94)(64,95)(97,179)(98,180)(99,181)(100,182)(101,183)(102,184)(103,185)(104,186)(105,187)(106,188)(107,189)(108,190)(109,191)(110,192)(111,161)(112,162)(113,163)(114,164)(115,165)(116,166)(117,167)(118,168)(119,169)(120,170)(121,171)(122,172)(123,173)(124,174)(125,175)(126,176)(127,177)(128,178), (1,44,109)(2,110,45)(3,46,111)(4,112,47)(5,48,113)(6,114,49)(7,50,115)(8,116,51)(9,52,117)(10,118,53)(11,54,119)(12,120,55)(13,56,121)(14,122,57)(15,58,123)(16,124,59)(17,60,125)(18,126,61)(19,62,127)(20,128,63)(21,64,97)(22,98,33)(23,34,99)(24,100,35)(25,36,101)(26,102,37)(27,38,103)(28,104,39)(29,40,105)(30,106,41)(31,42,107)(32,108,43)(65,181,138)(66,139,182)(67,183,140)(68,141,184)(69,185,142)(70,143,186)(71,187,144)(72,145,188)(73,189,146)(74,147,190)(75,191,148)(76,149,192)(77,161,150)(78,151,162)(79,163,152)(80,153,164)(81,165,154)(82,155,166)(83,167,156)(84,157,168)(85,169,158)(86,159,170)(87,171,160)(88,129,172)(89,173,130)(90,131,174)(91,175,132)(92,133,176)(93,177,134)(94,135,178)(95,179,136)(96,137,180), (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)>;
G:=Group( (1,148)(2,149)(3,150)(4,151)(5,152)(6,153)(7,154)(8,155)(9,156)(10,157)(11,158)(12,159)(13,160)(14,129)(15,130)(16,131)(17,132)(18,133)(19,134)(20,135)(21,136)(22,137)(23,138)(24,139)(25,140)(26,141)(27,142)(28,143)(29,144)(30,145)(31,146)(32,147)(33,96)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,71)(41,72)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79)(49,80)(50,81)(51,82)(52,83)(53,84)(54,85)(55,86)(56,87)(57,88)(58,89)(59,90)(60,91)(61,92)(62,93)(63,94)(64,95)(97,179)(98,180)(99,181)(100,182)(101,183)(102,184)(103,185)(104,186)(105,187)(106,188)(107,189)(108,190)(109,191)(110,192)(111,161)(112,162)(113,163)(114,164)(115,165)(116,166)(117,167)(118,168)(119,169)(120,170)(121,171)(122,172)(123,173)(124,174)(125,175)(126,176)(127,177)(128,178), (1,44,109)(2,110,45)(3,46,111)(4,112,47)(5,48,113)(6,114,49)(7,50,115)(8,116,51)(9,52,117)(10,118,53)(11,54,119)(12,120,55)(13,56,121)(14,122,57)(15,58,123)(16,124,59)(17,60,125)(18,126,61)(19,62,127)(20,128,63)(21,64,97)(22,98,33)(23,34,99)(24,100,35)(25,36,101)(26,102,37)(27,38,103)(28,104,39)(29,40,105)(30,106,41)(31,42,107)(32,108,43)(65,181,138)(66,139,182)(67,183,140)(68,141,184)(69,185,142)(70,143,186)(71,187,144)(72,145,188)(73,189,146)(74,147,190)(75,191,148)(76,149,192)(77,161,150)(78,151,162)(79,163,152)(80,153,164)(81,165,154)(82,155,166)(83,167,156)(84,157,168)(85,169,158)(86,159,170)(87,171,160)(88,129,172)(89,173,130)(90,131,174)(91,175,132)(92,133,176)(93,177,134)(94,135,178)(95,179,136)(96,137,180), (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) );
G=PermutationGroup([[(1,148),(2,149),(3,150),(4,151),(5,152),(6,153),(7,154),(8,155),(9,156),(10,157),(11,158),(12,159),(13,160),(14,129),(15,130),(16,131),(17,132),(18,133),(19,134),(20,135),(21,136),(22,137),(23,138),(24,139),(25,140),(26,141),(27,142),(28,143),(29,144),(30,145),(31,146),(32,147),(33,96),(34,65),(35,66),(36,67),(37,68),(38,69),(39,70),(40,71),(41,72),(42,73),(43,74),(44,75),(45,76),(46,77),(47,78),(48,79),(49,80),(50,81),(51,82),(52,83),(53,84),(54,85),(55,86),(56,87),(57,88),(58,89),(59,90),(60,91),(61,92),(62,93),(63,94),(64,95),(97,179),(98,180),(99,181),(100,182),(101,183),(102,184),(103,185),(104,186),(105,187),(106,188),(107,189),(108,190),(109,191),(110,192),(111,161),(112,162),(113,163),(114,164),(115,165),(116,166),(117,167),(118,168),(119,169),(120,170),(121,171),(122,172),(123,173),(124,174),(125,175),(126,176),(127,177),(128,178)], [(1,44,109),(2,110,45),(3,46,111),(4,112,47),(5,48,113),(6,114,49),(7,50,115),(8,116,51),(9,52,117),(10,118,53),(11,54,119),(12,120,55),(13,56,121),(14,122,57),(15,58,123),(16,124,59),(17,60,125),(18,126,61),(19,62,127),(20,128,63),(21,64,97),(22,98,33),(23,34,99),(24,100,35),(25,36,101),(26,102,37),(27,38,103),(28,104,39),(29,40,105),(30,106,41),(31,42,107),(32,108,43),(65,181,138),(66,139,182),(67,183,140),(68,141,184),(69,185,142),(70,143,186),(71,187,144),(72,145,188),(73,189,146),(74,147,190),(75,191,148),(76,149,192),(77,161,150),(78,151,162),(79,163,152),(80,153,164),(81,165,154),(82,155,166),(83,167,156),(84,157,168),(85,169,158),(86,159,170),(87,171,160),(88,129,172),(89,173,130),(90,131,174),(91,175,132),(92,133,176),(93,177,134),(94,135,178),(95,179,136),(96,137,180)], [(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)]])
96 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 8A | ··· | 8H | 12A | 12B | 12C | 12D | 16A | ··· | 16P | 24A | ··· | 24H | 32A | ··· | 32AF | 48A | ··· | 48P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 32 | ··· | 32 | 48 | ··· | 48 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | ||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | C16 | C32 | S3 | Dic3 | D6 | Dic3 | C3⋊C8 | C3⋊C8 | C3⋊C16 | C3⋊C16 | C3⋊C32 |
| kernel | C2×C3⋊C32 | C3⋊C32 | C2×C48 | C48 | C2×C24 | C24 | C2×C12 | C12 | C2×C6 | C6 | C2×C16 | C16 | C16 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 32 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 16 |
Matrix representation of C2×C3⋊C32 ►in GL4(𝔽97) generated by
| 1 | 0 | 0 | 0 |
| 0 | 96 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 96 | 96 |
| 0 | 0 | 1 | 0 |
| 67 | 0 | 0 | 0 |
| 0 | 96 | 0 | 0 |
| 0 | 0 | 8 | 26 |
| 0 | 0 | 18 | 89 |
G:=sub<GL(4,GF(97))| [1,0,0,0,0,96,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,96,1,0,0,96,0],[67,0,0,0,0,96,0,0,0,0,8,18,0,0,26,89] >;
C2×C3⋊C32 in GAP, Magma, Sage, TeX
C_2\times C_3\rtimes C_{32} % in TeX
G:=Group("C2xC3:C32"); // GroupNames label
G:=SmallGroup(192,57);
// by ID
G=gap.SmallGroup(192,57);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,58,80,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^2=b^3=c^32=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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