metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12⋊1(C4⋊C4), C6.57(C4×D4), C6.19(C4×Q8), C4⋊Dic3⋊12C4, C2.3(C12⋊Q8), C6.19(C4⋊Q8), (C2×C12).17Q8, C4⋊1(Dic3⋊C4), (C2×C12).136D4, (C2×C4).28Dic6, C2.3(D6⋊3D4), C22.18(S3×Q8), C6.87(C4⋊D4), C2.1(D6⋊3Q8), (C2×Dic3).14Q8, (C22×C4).347D6, C22.104(S3×D4), C6.69(C22⋊Q8), (C2×Dic3).107D4, C2.3(C4.Dic6), C6.15(C42.C2), C2.17(Dic3⋊5D4), C22.26(C2×Dic6), C2.9(Dic6⋊C4), C6.C42.14C2, (C22×C6).336C23, C23.296(C22×S3), C22.51(D4⋊2S3), (C22×C12).140C22, C3⋊3(C23.65C23), C22.21(Q8⋊3S3), (C22×Dic3).48C22, (C6×C4⋊C4).9C2, C6.35(C2×C4⋊C4), (C2×C4⋊C4).10S3, (C2×C4).77(C4×S3), (C2×C6).72(C2×Q8), (C2×C4×Dic3).3C2, (C2×C12).80(C2×C4), (C2×C6).327(C2×D4), C22.130(S3×C2×C4), C2.10(C2×Dic3⋊C4), (C2×C4⋊Dic3).31C2, C22.60(C2×C3⋊D4), (C2×C6).186(C4○D4), (C2×C4).183(C3⋊D4), (C2×Dic3⋊C4).11C2, (C2×C6).112(C22×C4), (C2×Dic3).28(C2×C4), SmallGroup(192,531)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12⋊(C4⋊C4)
G = < a,b,c | a12=b4=c4=1, bab-1=a-1, cac-1=a7, cbc-1=b-1 >
Subgroups: 376 in 170 conjugacy classes, 83 normal (41 characteristic)
C1, C2, C3, C4, C4, C22, C6, C2×C4, C2×C4, C23, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C2.C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C22×Dic3, C22×Dic3, C22×C12, C22×C12, C23.65C23, C6.C42, C2×C4×Dic3, C2×Dic3⋊C4, C2×C4⋊Dic3, C6×C4⋊C4, C12⋊(C4⋊C4)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic6, C4×S3, C3⋊D4, C22×S3, C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, Dic3⋊C4, C2×Dic6, S3×C2×C4, S3×D4, D4⋊2S3, S3×Q8, Q8⋊3S3, C2×C3⋊D4, C23.65C23, Dic6⋊C4, C12⋊Q8, C4.Dic6, Dic3⋊5D4, C2×Dic3⋊C4, D6⋊3D4, D6⋊3Q8, C12⋊(C4⋊C4)
►Regular action on 192 points
Generators in S192
(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)
(1 146 68 39)(2 145 69 38)(3 156 70 37)(4 155 71 48)(5 154 72 47)(6 153 61 46)(7 152 62 45)(8 151 63 44)(9 150 64 43)(10 149 65 42)(11 148 66 41)(12 147 67 40)(13 109 144 30)(14 120 133 29)(15 119 134 28)(16 118 135 27)(17 117 136 26)(18 116 137 25)(19 115 138 36)(20 114 139 35)(21 113 140 34)(22 112 141 33)(23 111 142 32)(24 110 143 31)(49 171 80 124)(50 170 81 123)(51 169 82 122)(52 180 83 121)(53 179 84 132)(54 178 73 131)(55 177 74 130)(56 176 75 129)(57 175 76 128)(58 174 77 127)(59 173 78 126)(60 172 79 125)(85 107 162 189)(86 106 163 188)(87 105 164 187)(88 104 165 186)(89 103 166 185)(90 102 167 184)(91 101 168 183)(92 100 157 182)(93 99 158 181)(94 98 159 192)(95 97 160 191)(96 108 161 190)
(1 125 95 34)(2 132 96 29)(3 127 85 36)(4 122 86 31)(5 129 87 26)(6 124 88 33)(7 131 89 28)(8 126 90 35)(9 121 91 30)(10 128 92 25)(11 123 93 32)(12 130 94 27)(13 43 52 183)(14 38 53 190)(15 45 54 185)(16 40 55 192)(17 47 56 187)(18 42 57 182)(19 37 58 189)(20 44 59 184)(21 39 60 191)(22 46 49 186)(23 41 50 181)(24 48 51 188)(61 171 165 112)(62 178 166 119)(63 173 167 114)(64 180 168 109)(65 175 157 116)(66 170 158 111)(67 177 159 118)(68 172 160 113)(69 179 161 120)(70 174 162 115)(71 169 163 110)(72 176 164 117)(73 103 134 152)(74 98 135 147)(75 105 136 154)(76 100 137 149)(77 107 138 156)(78 102 139 151)(79 97 140 146)(80 104 141 153)(81 99 142 148)(82 106 143 155)(83 101 144 150)(84 108 133 145)
G:=sub<Sym(192)| (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), (1,146,68,39)(2,145,69,38)(3,156,70,37)(4,155,71,48)(5,154,72,47)(6,153,61,46)(7,152,62,45)(8,151,63,44)(9,150,64,43)(10,149,65,42)(11,148,66,41)(12,147,67,40)(13,109,144,30)(14,120,133,29)(15,119,134,28)(16,118,135,27)(17,117,136,26)(18,116,137,25)(19,115,138,36)(20,114,139,35)(21,113,140,34)(22,112,141,33)(23,111,142,32)(24,110,143,31)(49,171,80,124)(50,170,81,123)(51,169,82,122)(52,180,83,121)(53,179,84,132)(54,178,73,131)(55,177,74,130)(56,176,75,129)(57,175,76,128)(58,174,77,127)(59,173,78,126)(60,172,79,125)(85,107,162,189)(86,106,163,188)(87,105,164,187)(88,104,165,186)(89,103,166,185)(90,102,167,184)(91,101,168,183)(92,100,157,182)(93,99,158,181)(94,98,159,192)(95,97,160,191)(96,108,161,190), (1,125,95,34)(2,132,96,29)(3,127,85,36)(4,122,86,31)(5,129,87,26)(6,124,88,33)(7,131,89,28)(8,126,90,35)(9,121,91,30)(10,128,92,25)(11,123,93,32)(12,130,94,27)(13,43,52,183)(14,38,53,190)(15,45,54,185)(16,40,55,192)(17,47,56,187)(18,42,57,182)(19,37,58,189)(20,44,59,184)(21,39,60,191)(22,46,49,186)(23,41,50,181)(24,48,51,188)(61,171,165,112)(62,178,166,119)(63,173,167,114)(64,180,168,109)(65,175,157,116)(66,170,158,111)(67,177,159,118)(68,172,160,113)(69,179,161,120)(70,174,162,115)(71,169,163,110)(72,176,164,117)(73,103,134,152)(74,98,135,147)(75,105,136,154)(76,100,137,149)(77,107,138,156)(78,102,139,151)(79,97,140,146)(80,104,141,153)(81,99,142,148)(82,106,143,155)(83,101,144,150)(84,108,133,145)>;
G:=Group( (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), (1,146,68,39)(2,145,69,38)(3,156,70,37)(4,155,71,48)(5,154,72,47)(6,153,61,46)(7,152,62,45)(8,151,63,44)(9,150,64,43)(10,149,65,42)(11,148,66,41)(12,147,67,40)(13,109,144,30)(14,120,133,29)(15,119,134,28)(16,118,135,27)(17,117,136,26)(18,116,137,25)(19,115,138,36)(20,114,139,35)(21,113,140,34)(22,112,141,33)(23,111,142,32)(24,110,143,31)(49,171,80,124)(50,170,81,123)(51,169,82,122)(52,180,83,121)(53,179,84,132)(54,178,73,131)(55,177,74,130)(56,176,75,129)(57,175,76,128)(58,174,77,127)(59,173,78,126)(60,172,79,125)(85,107,162,189)(86,106,163,188)(87,105,164,187)(88,104,165,186)(89,103,166,185)(90,102,167,184)(91,101,168,183)(92,100,157,182)(93,99,158,181)(94,98,159,192)(95,97,160,191)(96,108,161,190), (1,125,95,34)(2,132,96,29)(3,127,85,36)(4,122,86,31)(5,129,87,26)(6,124,88,33)(7,131,89,28)(8,126,90,35)(9,121,91,30)(10,128,92,25)(11,123,93,32)(12,130,94,27)(13,43,52,183)(14,38,53,190)(15,45,54,185)(16,40,55,192)(17,47,56,187)(18,42,57,182)(19,37,58,189)(20,44,59,184)(21,39,60,191)(22,46,49,186)(23,41,50,181)(24,48,51,188)(61,171,165,112)(62,178,166,119)(63,173,167,114)(64,180,168,109)(65,175,157,116)(66,170,158,111)(67,177,159,118)(68,172,160,113)(69,179,161,120)(70,174,162,115)(71,169,163,110)(72,176,164,117)(73,103,134,152)(74,98,135,147)(75,105,136,154)(76,100,137,149)(77,107,138,156)(78,102,139,151)(79,97,140,146)(80,104,141,153)(81,99,142,148)(82,106,143,155)(83,101,144,150)(84,108,133,145) );
G=PermutationGroup([[(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)], [(1,146,68,39),(2,145,69,38),(3,156,70,37),(4,155,71,48),(5,154,72,47),(6,153,61,46),(7,152,62,45),(8,151,63,44),(9,150,64,43),(10,149,65,42),(11,148,66,41),(12,147,67,40),(13,109,144,30),(14,120,133,29),(15,119,134,28),(16,118,135,27),(17,117,136,26),(18,116,137,25),(19,115,138,36),(20,114,139,35),(21,113,140,34),(22,112,141,33),(23,111,142,32),(24,110,143,31),(49,171,80,124),(50,170,81,123),(51,169,82,122),(52,180,83,121),(53,179,84,132),(54,178,73,131),(55,177,74,130),(56,176,75,129),(57,175,76,128),(58,174,77,127),(59,173,78,126),(60,172,79,125),(85,107,162,189),(86,106,163,188),(87,105,164,187),(88,104,165,186),(89,103,166,185),(90,102,167,184),(91,101,168,183),(92,100,157,182),(93,99,158,181),(94,98,159,192),(95,97,160,191),(96,108,161,190)], [(1,125,95,34),(2,132,96,29),(3,127,85,36),(4,122,86,31),(5,129,87,26),(6,124,88,33),(7,131,89,28),(8,126,90,35),(9,121,91,30),(10,128,92,25),(11,123,93,32),(12,130,94,27),(13,43,52,183),(14,38,53,190),(15,45,54,185),(16,40,55,192),(17,47,56,187),(18,42,57,182),(19,37,58,189),(20,44,59,184),(21,39,60,191),(22,46,49,186),(23,41,50,181),(24,48,51,188),(61,171,165,112),(62,178,166,119),(63,173,167,114),(64,180,168,109),(65,175,157,116),(66,170,158,111),(67,177,159,118),(68,172,160,113),(69,179,161,120),(70,174,162,115),(71,169,163,110),(72,176,164,117),(73,103,134,152),(74,98,135,147),(75,105,136,154),(76,100,137,149),(77,107,138,156),(78,102,139,151),(79,97,140,146),(80,104,141,153),(81,99,142,148),(82,106,143,155),(83,101,144,150),(84,108,133,145)]])
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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | - | + | - | + | - | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | Q8 | D4 | Q8 | D6 | C4○D4 | Dic6 | C4×S3 | C3⋊D4 | S3×D4 | D4⋊2S3 | S3×Q8 | Q8⋊3S3 |
| kernel | C12⋊(C4⋊C4) | C6.C42 | C2×C4×Dic3 | C2×Dic3⋊C4 | C2×C4⋊Dic3 | C6×C4⋊C4 | C4⋊Dic3 | C2×C4⋊C4 | C2×Dic3 | C2×Dic3 | C2×C12 | C2×C12 | C22×C4 | C2×C6 | C2×C4 | C2×C4 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 8 | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 |
Matrix representation of C12⋊(C4⋊C4) ►in GL7(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 11 | 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 12 | 0 | 0 |
| 0 | 0 | 0 | 10 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(7,GF(13))| [12,0,0,0,0,0,0,0,12,1,0,0,0,0,0,11,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,12,0],[1,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,10,0,0,0,0,0,12,10,0,0,0,0,0,0,0,0,12,0,0,0,0,0,12,0],[5,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,11,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1] >;
C12⋊(C4⋊C4) in GAP, Magma, Sage, TeX
C_{12}\rtimes (C_4\rtimes C_4) % in TeX
G:=Group("C12:(C4:C4)"); // GroupNames label
G:=SmallGroup(192,531);
// by ID
G=gap.SmallGroup(192,531);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,120,254,219,184,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^4=c^4=1,b*a*b^-1=a^-1,c*a*c^-1=a^7,c*b*c^-1=b^-1>;
// generators/relations