direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×C4⋊Dic3, C12⋊2C42, C42⋊9Dic3, C12⋊7(C4⋊C4), (C4×C12)⋊13C4, C4⋊2(C4×Dic3), C6.16(C4×D4), C2.2(C4×D12), C6.12(C4×Q8), (C2×C12).67Q8, C2.3(C4×Dic6), (C2×C4).167D12, (C2×C12).400D4, (C2×C42).16S3, C6.18(C2×C42), (C2×C4).57Dic6, (C22×C4).434D6, C22.35(C2×D12), C22.20(C2×Dic6), C6.41(C42⋊C2), C22.44(C4○D12), C6.C42.43C2, (C22×C6).309C23, C23.277(C22×S3), (C22×C12).474C22, C2.3(C23.26D6), C22.18(C22×Dic3), (C22×Dic3).180C22, C3⋊3(C4×C4⋊C4), C6.26(C2×C4⋊C4), (C2×C4×C12).21C2, C2.7(C2×C4×Dic3), C22.52(S3×C2×C4), C2.2(C2×C4⋊Dic3), (C2×C6).27(C2×Q8), (C2×C4).110(C4×S3), (C2×C6).145(C2×D4), (C2×C4×Dic3).33C2, (C2×C12).317(C2×C4), (C2×C6).69(C4○D4), (C2×C4⋊Dic3).46C2, (C2×C6).98(C22×C4), (C2×C4).61(C2×Dic3), (C2×Dic3).57(C2×C4), SmallGroup(192,493)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×C4⋊Dic3
G = < a,b,c,d | a4=b4=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 376 in 194 conjugacy classes, 119 normal (25 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C42, C4⋊C4, C22×C4, C22×C4, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C4×Dic3, C4⋊Dic3, C4×C12, C22×Dic3, C22×C12, C4×C4⋊C4, C6.C42, C2×C4×Dic3, C2×C4⋊Dic3, C2×C4×C12, C4×C4⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, Dic3, D6, C42, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic6, C4×S3, D12, C2×Dic3, C22×S3, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C4×Dic3, C4⋊Dic3, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C22×Dic3, C4×C4⋊C4, C4×Dic6, C4×D12, C2×C4×Dic3, C2×C4⋊Dic3, C23.26D6, C4×C4⋊Dic3
►Regular action on 192 points
Generators in S192
(1 40 9 35)(2 41 10 36)(3 42 11 31)(4 37 12 32)(5 38 7 33)(6 39 8 34)(13 172 191 167)(14 173 192 168)(15 174 187 163)(16 169 188 164)(17 170 189 165)(18 171 190 166)(19 53 29 43)(20 54 30 44)(21 49 25 45)(22 50 26 46)(23 51 27 47)(24 52 28 48)(55 90 66 79)(56 85 61 80)(57 86 62 81)(58 87 63 82)(59 88 64 83)(60 89 65 84)(67 102 77 91)(68 97 78 92)(69 98 73 93)(70 99 74 94)(71 100 75 95)(72 101 76 96)(103 137 113 127)(104 138 114 128)(105 133 109 129)(106 134 110 130)(107 135 111 131)(108 136 112 132)(115 150 126 139)(116 145 121 140)(117 146 122 141)(118 147 123 142)(119 148 124 143)(120 149 125 144)(151 185 161 175)(152 186 162 176)(153 181 157 177)(154 182 158 178)(155 183 159 179)(156 184 160 180)
(1 67 19 59)(2 68 20 60)(3 69 21 55)(4 70 22 56)(5 71 23 57)(6 72 24 58)(7 75 27 62)(8 76 28 63)(9 77 29 64)(10 78 30 65)(11 73 25 66)(12 74 26 61)(13 134 182 148)(14 135 183 149)(15 136 184 150)(16 137 185 145)(17 138 186 146)(18 133 181 147)(31 93 45 79)(32 94 46 80)(33 95 47 81)(34 96 48 82)(35 91 43 83)(36 92 44 84)(37 99 50 85)(38 100 51 86)(39 101 52 87)(40 102 53 88)(41 97 54 89)(42 98 49 90)(103 151 116 164)(104 152 117 165)(105 153 118 166)(106 154 119 167)(107 155 120 168)(108 156 115 163)(109 157 123 171)(110 158 124 172)(111 159 125 173)(112 160 126 174)(113 161 121 169)(114 162 122 170)(127 175 140 188)(128 176 141 189)(129 177 142 190)(130 178 143 191)(131 179 144 192)(132 180 139 187)
(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)
(1 103 4 106)(2 108 5 105)(3 107 6 104)(7 109 10 112)(8 114 11 111)(9 113 12 110)(13 102 16 99)(14 101 17 98)(15 100 18 97)(19 116 22 119)(20 115 23 118)(21 120 24 117)(25 125 28 122)(26 124 29 121)(27 123 30 126)(31 131 34 128)(32 130 35 127)(33 129 36 132)(37 134 40 137)(38 133 41 136)(39 138 42 135)(43 140 46 143)(44 139 47 142)(45 144 48 141)(49 149 52 146)(50 148 53 145)(51 147 54 150)(55 155 58 152)(56 154 59 151)(57 153 60 156)(61 158 64 161)(62 157 65 160)(63 162 66 159)(67 164 70 167)(68 163 71 166)(69 168 72 165)(73 173 76 170)(74 172 77 169)(75 171 78 174)(79 179 82 176)(80 178 83 175)(81 177 84 180)(85 182 88 185)(86 181 89 184)(87 186 90 183)(91 188 94 191)(92 187 95 190)(93 192 96 189)
G:=sub<Sym(192)| (1,40,9,35)(2,41,10,36)(3,42,11,31)(4,37,12,32)(5,38,7,33)(6,39,8,34)(13,172,191,167)(14,173,192,168)(15,174,187,163)(16,169,188,164)(17,170,189,165)(18,171,190,166)(19,53,29,43)(20,54,30,44)(21,49,25,45)(22,50,26,46)(23,51,27,47)(24,52,28,48)(55,90,66,79)(56,85,61,80)(57,86,62,81)(58,87,63,82)(59,88,64,83)(60,89,65,84)(67,102,77,91)(68,97,78,92)(69,98,73,93)(70,99,74,94)(71,100,75,95)(72,101,76,96)(103,137,113,127)(104,138,114,128)(105,133,109,129)(106,134,110,130)(107,135,111,131)(108,136,112,132)(115,150,126,139)(116,145,121,140)(117,146,122,141)(118,147,123,142)(119,148,124,143)(120,149,125,144)(151,185,161,175)(152,186,162,176)(153,181,157,177)(154,182,158,178)(155,183,159,179)(156,184,160,180), (1,67,19,59)(2,68,20,60)(3,69,21,55)(4,70,22,56)(5,71,23,57)(6,72,24,58)(7,75,27,62)(8,76,28,63)(9,77,29,64)(10,78,30,65)(11,73,25,66)(12,74,26,61)(13,134,182,148)(14,135,183,149)(15,136,184,150)(16,137,185,145)(17,138,186,146)(18,133,181,147)(31,93,45,79)(32,94,46,80)(33,95,47,81)(34,96,48,82)(35,91,43,83)(36,92,44,84)(37,99,50,85)(38,100,51,86)(39,101,52,87)(40,102,53,88)(41,97,54,89)(42,98,49,90)(103,151,116,164)(104,152,117,165)(105,153,118,166)(106,154,119,167)(107,155,120,168)(108,156,115,163)(109,157,123,171)(110,158,124,172)(111,159,125,173)(112,160,126,174)(113,161,121,169)(114,162,122,170)(127,175,140,188)(128,176,141,189)(129,177,142,190)(130,178,143,191)(131,179,144,192)(132,180,139,187), (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), (1,103,4,106)(2,108,5,105)(3,107,6,104)(7,109,10,112)(8,114,11,111)(9,113,12,110)(13,102,16,99)(14,101,17,98)(15,100,18,97)(19,116,22,119)(20,115,23,118)(21,120,24,117)(25,125,28,122)(26,124,29,121)(27,123,30,126)(31,131,34,128)(32,130,35,127)(33,129,36,132)(37,134,40,137)(38,133,41,136)(39,138,42,135)(43,140,46,143)(44,139,47,142)(45,144,48,141)(49,149,52,146)(50,148,53,145)(51,147,54,150)(55,155,58,152)(56,154,59,151)(57,153,60,156)(61,158,64,161)(62,157,65,160)(63,162,66,159)(67,164,70,167)(68,163,71,166)(69,168,72,165)(73,173,76,170)(74,172,77,169)(75,171,78,174)(79,179,82,176)(80,178,83,175)(81,177,84,180)(85,182,88,185)(86,181,89,184)(87,186,90,183)(91,188,94,191)(92,187,95,190)(93,192,96,189)>;
G:=Group( (1,40,9,35)(2,41,10,36)(3,42,11,31)(4,37,12,32)(5,38,7,33)(6,39,8,34)(13,172,191,167)(14,173,192,168)(15,174,187,163)(16,169,188,164)(17,170,189,165)(18,171,190,166)(19,53,29,43)(20,54,30,44)(21,49,25,45)(22,50,26,46)(23,51,27,47)(24,52,28,48)(55,90,66,79)(56,85,61,80)(57,86,62,81)(58,87,63,82)(59,88,64,83)(60,89,65,84)(67,102,77,91)(68,97,78,92)(69,98,73,93)(70,99,74,94)(71,100,75,95)(72,101,76,96)(103,137,113,127)(104,138,114,128)(105,133,109,129)(106,134,110,130)(107,135,111,131)(108,136,112,132)(115,150,126,139)(116,145,121,140)(117,146,122,141)(118,147,123,142)(119,148,124,143)(120,149,125,144)(151,185,161,175)(152,186,162,176)(153,181,157,177)(154,182,158,178)(155,183,159,179)(156,184,160,180), (1,67,19,59)(2,68,20,60)(3,69,21,55)(4,70,22,56)(5,71,23,57)(6,72,24,58)(7,75,27,62)(8,76,28,63)(9,77,29,64)(10,78,30,65)(11,73,25,66)(12,74,26,61)(13,134,182,148)(14,135,183,149)(15,136,184,150)(16,137,185,145)(17,138,186,146)(18,133,181,147)(31,93,45,79)(32,94,46,80)(33,95,47,81)(34,96,48,82)(35,91,43,83)(36,92,44,84)(37,99,50,85)(38,100,51,86)(39,101,52,87)(40,102,53,88)(41,97,54,89)(42,98,49,90)(103,151,116,164)(104,152,117,165)(105,153,118,166)(106,154,119,167)(107,155,120,168)(108,156,115,163)(109,157,123,171)(110,158,124,172)(111,159,125,173)(112,160,126,174)(113,161,121,169)(114,162,122,170)(127,175,140,188)(128,176,141,189)(129,177,142,190)(130,178,143,191)(131,179,144,192)(132,180,139,187), (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), (1,103,4,106)(2,108,5,105)(3,107,6,104)(7,109,10,112)(8,114,11,111)(9,113,12,110)(13,102,16,99)(14,101,17,98)(15,100,18,97)(19,116,22,119)(20,115,23,118)(21,120,24,117)(25,125,28,122)(26,124,29,121)(27,123,30,126)(31,131,34,128)(32,130,35,127)(33,129,36,132)(37,134,40,137)(38,133,41,136)(39,138,42,135)(43,140,46,143)(44,139,47,142)(45,144,48,141)(49,149,52,146)(50,148,53,145)(51,147,54,150)(55,155,58,152)(56,154,59,151)(57,153,60,156)(61,158,64,161)(62,157,65,160)(63,162,66,159)(67,164,70,167)(68,163,71,166)(69,168,72,165)(73,173,76,170)(74,172,77,169)(75,171,78,174)(79,179,82,176)(80,178,83,175)(81,177,84,180)(85,182,88,185)(86,181,89,184)(87,186,90,183)(91,188,94,191)(92,187,95,190)(93,192,96,189) );
G=PermutationGroup([[(1,40,9,35),(2,41,10,36),(3,42,11,31),(4,37,12,32),(5,38,7,33),(6,39,8,34),(13,172,191,167),(14,173,192,168),(15,174,187,163),(16,169,188,164),(17,170,189,165),(18,171,190,166),(19,53,29,43),(20,54,30,44),(21,49,25,45),(22,50,26,46),(23,51,27,47),(24,52,28,48),(55,90,66,79),(56,85,61,80),(57,86,62,81),(58,87,63,82),(59,88,64,83),(60,89,65,84),(67,102,77,91),(68,97,78,92),(69,98,73,93),(70,99,74,94),(71,100,75,95),(72,101,76,96),(103,137,113,127),(104,138,114,128),(105,133,109,129),(106,134,110,130),(107,135,111,131),(108,136,112,132),(115,150,126,139),(116,145,121,140),(117,146,122,141),(118,147,123,142),(119,148,124,143),(120,149,125,144),(151,185,161,175),(152,186,162,176),(153,181,157,177),(154,182,158,178),(155,183,159,179),(156,184,160,180)], [(1,67,19,59),(2,68,20,60),(3,69,21,55),(4,70,22,56),(5,71,23,57),(6,72,24,58),(7,75,27,62),(8,76,28,63),(9,77,29,64),(10,78,30,65),(11,73,25,66),(12,74,26,61),(13,134,182,148),(14,135,183,149),(15,136,184,150),(16,137,185,145),(17,138,186,146),(18,133,181,147),(31,93,45,79),(32,94,46,80),(33,95,47,81),(34,96,48,82),(35,91,43,83),(36,92,44,84),(37,99,50,85),(38,100,51,86),(39,101,52,87),(40,102,53,88),(41,97,54,89),(42,98,49,90),(103,151,116,164),(104,152,117,165),(105,153,118,166),(106,154,119,167),(107,155,120,168),(108,156,115,163),(109,157,123,171),(110,158,124,172),(111,159,125,173),(112,160,126,174),(113,161,121,169),(114,162,122,170),(127,175,140,188),(128,176,141,189),(129,177,142,190),(130,178,143,191),(131,179,144,192),(132,180,139,187)], [(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)], [(1,103,4,106),(2,108,5,105),(3,107,6,104),(7,109,10,112),(8,114,11,111),(9,113,12,110),(13,102,16,99),(14,101,17,98),(15,100,18,97),(19,116,22,119),(20,115,23,118),(21,120,24,117),(25,125,28,122),(26,124,29,121),(27,123,30,126),(31,131,34,128),(32,130,35,127),(33,129,36,132),(37,134,40,137),(38,133,41,136),(39,138,42,135),(43,140,46,143),(44,139,47,142),(45,144,48,141),(49,149,52,146),(50,148,53,145),(51,147,54,150),(55,155,58,152),(56,154,59,151),(57,153,60,156),(61,158,64,161),(62,157,65,160),(63,162,66,159),(67,164,70,167),(68,163,71,166),(69,168,72,165),(73,173,76,170),(74,172,77,169),(75,171,78,174),(79,179,82,176),(80,178,83,175),(81,177,84,180),(85,182,88,185),(86,181,89,184),(87,186,90,183),(91,188,94,191),(92,187,95,190),(93,192,96,189)]])
72 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3 | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | ··· | 4AF | 6A | ··· | 6G | 12A | ··· | 12X |
| order | 1 | 2 | ··· | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | - | + | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | Q8 | Dic3 | D6 | C4○D4 | Dic6 | C4×S3 | D12 | C4○D12 |
| kernel | C4×C4⋊Dic3 | C6.C42 | C2×C4×Dic3 | C2×C4⋊Dic3 | C2×C4×C12 | C4⋊Dic3 | C4×C12 | C2×C42 | C2×C12 | C2×C12 | C42 | C22×C4 | C2×C6 | C2×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 2 | 2 | 2 | 1 | 16 | 8 | 1 | 2 | 2 | 4 | 3 | 4 | 4 | 8 | 4 | 8 |
Matrix representation of C4×C4⋊Dic3 ►in GL5(𝔽13)
| 8 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 5 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 | 0 |
| 0 | 3 | 4 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 5 | 0 | 0 | 0 | 0 |
| 0 | 10 | 6 | 0 | 0 |
| 0 | 7 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(13))| [8,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,8,0,0,0,0,0,5],[12,0,0,0,0,0,10,3,0,0,0,0,4,0,0,0,0,0,9,0,0,0,0,0,3],[5,0,0,0,0,0,10,7,0,0,0,6,3,0,0,0,0,0,0,1,0,0,0,1,0] >;
C4×C4⋊Dic3 in GAP, Magma, Sage, TeX
C_4\times C_4\rtimes {\rm Dic}_3 % in TeX
G:=Group("C4xC4:Dic3"); // GroupNames label
G:=SmallGroup(192,493);
// by ID
G=gap.SmallGroup(192,493);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,477,232,100,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations