metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12⋊4(C4⋊C4), C6.8(C4×Q8), C4⋊Dic3⋊9C4, C6.15(C4×D4), C6.6(C4⋊Q8), C2.18(C4×D12), (C2×C4).65D12, (C2×C12).52Q8, C4⋊3(Dic3⋊C4), (C2×C12).469D4, (C2×C42).14S3, C2.10(C4×Dic6), (C2×C4).40Dic6, C6.55(C4⋊D4), C2.1(C12⋊7D4), C2.1(C12⋊2Q8), (C22×C4).410D6, C22.33(C2×D12), C6.52(C22⋊Q8), C6.2(C42.C2), C6.C42.9C2, C2.1(C12.6Q8), C22.16(C2×Dic6), C2.1(C12.48D4), C22.39(C4○D12), C23.271(C22×S3), (C22×C6).303C23, (C22×C12).469C22, C3⋊2(C23.65C23), (C22×Dic3).26C22, (C2×C4×C12).9C2, C6.24(C2×C4⋊C4), (C2×C6).23(C2×Q8), (C2×C4).108(C4×S3), (C2×C6).423(C2×D4), C2.4(C2×Dic3⋊C4), C22.116(S3×C2×C4), (C2×C12).223(C2×C4), (C2×C6).64(C4○D4), (C2×Dic3⋊C4).9C2, (C2×C4⋊Dic3).13C2, (C2×C6).93(C22×C4), C22.38(C2×C3⋊D4), (C2×C4).237(C3⋊D4), (C2×Dic3).24(C2×C4), SmallGroup(192,487)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12⋊4(C4⋊C4)
G = < a,b,c | a12=b4=c4=1, bab-1=a-1, ac=ca, cbc-1=b-1 >
Subgroups: 376 in 170 conjugacy classes, 87 normal (43 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, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C2.C42, C2×C42, C2×C4⋊C4, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C4×C12, C22×Dic3, C22×C12, C23.65C23, C6.C42, C2×Dic3⋊C4, C2×C4⋊Dic3, C2×C4×C12, C12⋊4(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, D12, 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, C2×D12, C4○D12, C2×C3⋊D4, C23.65C23, C4×Dic6, C12⋊2Q8, C12.6Q8, C4×D12, C2×Dic3⋊C4, C12.48D4, C12⋊7D4, C12⋊4(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)
(1 136 160 190)(2 135 161 189)(3 134 162 188)(4 133 163 187)(5 144 164 186)(6 143 165 185)(7 142 166 184)(8 141 167 183)(9 140 168 182)(10 139 157 181)(11 138 158 192)(12 137 159 191)(13 25 146 107)(14 36 147 106)(15 35 148 105)(16 34 149 104)(17 33 150 103)(18 32 151 102)(19 31 152 101)(20 30 153 100)(21 29 154 99)(22 28 155 98)(23 27 156 97)(24 26 145 108)(37 89 71 120)(38 88 72 119)(39 87 61 118)(40 86 62 117)(41 85 63 116)(42 96 64 115)(43 95 65 114)(44 94 66 113)(45 93 67 112)(46 92 68 111)(47 91 69 110)(48 90 70 109)(49 76 130 169)(50 75 131 180)(51 74 132 179)(52 73 121 178)(53 84 122 177)(54 83 123 176)(55 82 124 175)(56 81 125 174)(57 80 126 173)(58 79 127 172)(59 78 128 171)(60 77 129 170)
(1 39 128 155)(2 40 129 156)(3 41 130 145)(4 42 131 146)(5 43 132 147)(6 44 121 148)(7 45 122 149)(8 46 123 150)(9 47 124 151)(10 48 125 152)(11 37 126 153)(12 38 127 154)(13 163 64 50)(14 164 65 51)(15 165 66 52)(16 166 67 53)(17 167 68 54)(18 168 69 55)(19 157 70 56)(20 158 71 57)(21 159 72 58)(22 160 61 59)(23 161 62 60)(24 162 63 49)(25 133 115 180)(26 134 116 169)(27 135 117 170)(28 136 118 171)(29 137 119 172)(30 138 120 173)(31 139 109 174)(32 140 110 175)(33 141 111 176)(34 142 112 177)(35 143 113 178)(36 144 114 179)(73 105 185 94)(74 106 186 95)(75 107 187 96)(76 108 188 85)(77 97 189 86)(78 98 190 87)(79 99 191 88)(80 100 192 89)(81 101 181 90)(82 102 182 91)(83 103 183 92)(84 104 184 93)
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), (1,136,160,190)(2,135,161,189)(3,134,162,188)(4,133,163,187)(5,144,164,186)(6,143,165,185)(7,142,166,184)(8,141,167,183)(9,140,168,182)(10,139,157,181)(11,138,158,192)(12,137,159,191)(13,25,146,107)(14,36,147,106)(15,35,148,105)(16,34,149,104)(17,33,150,103)(18,32,151,102)(19,31,152,101)(20,30,153,100)(21,29,154,99)(22,28,155,98)(23,27,156,97)(24,26,145,108)(37,89,71,120)(38,88,72,119)(39,87,61,118)(40,86,62,117)(41,85,63,116)(42,96,64,115)(43,95,65,114)(44,94,66,113)(45,93,67,112)(46,92,68,111)(47,91,69,110)(48,90,70,109)(49,76,130,169)(50,75,131,180)(51,74,132,179)(52,73,121,178)(53,84,122,177)(54,83,123,176)(55,82,124,175)(56,81,125,174)(57,80,126,173)(58,79,127,172)(59,78,128,171)(60,77,129,170), (1,39,128,155)(2,40,129,156)(3,41,130,145)(4,42,131,146)(5,43,132,147)(6,44,121,148)(7,45,122,149)(8,46,123,150)(9,47,124,151)(10,48,125,152)(11,37,126,153)(12,38,127,154)(13,163,64,50)(14,164,65,51)(15,165,66,52)(16,166,67,53)(17,167,68,54)(18,168,69,55)(19,157,70,56)(20,158,71,57)(21,159,72,58)(22,160,61,59)(23,161,62,60)(24,162,63,49)(25,133,115,180)(26,134,116,169)(27,135,117,170)(28,136,118,171)(29,137,119,172)(30,138,120,173)(31,139,109,174)(32,140,110,175)(33,141,111,176)(34,142,112,177)(35,143,113,178)(36,144,114,179)(73,105,185,94)(74,106,186,95)(75,107,187,96)(76,108,188,85)(77,97,189,86)(78,98,190,87)(79,99,191,88)(80,100,192,89)(81,101,181,90)(82,102,182,91)(83,103,183,92)(84,104,184,93)>;
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), (1,136,160,190)(2,135,161,189)(3,134,162,188)(4,133,163,187)(5,144,164,186)(6,143,165,185)(7,142,166,184)(8,141,167,183)(9,140,168,182)(10,139,157,181)(11,138,158,192)(12,137,159,191)(13,25,146,107)(14,36,147,106)(15,35,148,105)(16,34,149,104)(17,33,150,103)(18,32,151,102)(19,31,152,101)(20,30,153,100)(21,29,154,99)(22,28,155,98)(23,27,156,97)(24,26,145,108)(37,89,71,120)(38,88,72,119)(39,87,61,118)(40,86,62,117)(41,85,63,116)(42,96,64,115)(43,95,65,114)(44,94,66,113)(45,93,67,112)(46,92,68,111)(47,91,69,110)(48,90,70,109)(49,76,130,169)(50,75,131,180)(51,74,132,179)(52,73,121,178)(53,84,122,177)(54,83,123,176)(55,82,124,175)(56,81,125,174)(57,80,126,173)(58,79,127,172)(59,78,128,171)(60,77,129,170), (1,39,128,155)(2,40,129,156)(3,41,130,145)(4,42,131,146)(5,43,132,147)(6,44,121,148)(7,45,122,149)(8,46,123,150)(9,47,124,151)(10,48,125,152)(11,37,126,153)(12,38,127,154)(13,163,64,50)(14,164,65,51)(15,165,66,52)(16,166,67,53)(17,167,68,54)(18,168,69,55)(19,157,70,56)(20,158,71,57)(21,159,72,58)(22,160,61,59)(23,161,62,60)(24,162,63,49)(25,133,115,180)(26,134,116,169)(27,135,117,170)(28,136,118,171)(29,137,119,172)(30,138,120,173)(31,139,109,174)(32,140,110,175)(33,141,111,176)(34,142,112,177)(35,143,113,178)(36,144,114,179)(73,105,185,94)(74,106,186,95)(75,107,187,96)(76,108,188,85)(77,97,189,86)(78,98,190,87)(79,99,191,88)(80,100,192,89)(81,101,181,90)(82,102,182,91)(83,103,183,92)(84,104,184,93) );
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)], [(1,136,160,190),(2,135,161,189),(3,134,162,188),(4,133,163,187),(5,144,164,186),(6,143,165,185),(7,142,166,184),(8,141,167,183),(9,140,168,182),(10,139,157,181),(11,138,158,192),(12,137,159,191),(13,25,146,107),(14,36,147,106),(15,35,148,105),(16,34,149,104),(17,33,150,103),(18,32,151,102),(19,31,152,101),(20,30,153,100),(21,29,154,99),(22,28,155,98),(23,27,156,97),(24,26,145,108),(37,89,71,120),(38,88,72,119),(39,87,61,118),(40,86,62,117),(41,85,63,116),(42,96,64,115),(43,95,65,114),(44,94,66,113),(45,93,67,112),(46,92,68,111),(47,91,69,110),(48,90,70,109),(49,76,130,169),(50,75,131,180),(51,74,132,179),(52,73,121,178),(53,84,122,177),(54,83,123,176),(55,82,124,175),(56,81,125,174),(57,80,126,173),(58,79,127,172),(59,78,128,171),(60,77,129,170)], [(1,39,128,155),(2,40,129,156),(3,41,130,145),(4,42,131,146),(5,43,132,147),(6,44,121,148),(7,45,122,149),(8,46,123,150),(9,47,124,151),(10,48,125,152),(11,37,126,153),(12,38,127,154),(13,163,64,50),(14,164,65,51),(15,165,66,52),(16,166,67,53),(17,167,68,54),(18,168,69,55),(19,157,70,56),(20,158,71,57),(21,159,72,58),(22,160,61,59),(23,161,62,60),(24,162,63,49),(25,133,115,180),(26,134,116,169),(27,135,117,170),(28,136,118,171),(29,137,119,172),(30,138,120,173),(31,139,109,174),(32,140,110,175),(33,141,111,176),(34,142,112,177),(35,143,113,178),(36,144,114,179),(73,105,185,94),(74,106,186,95),(75,107,187,96),(76,108,188,85),(77,97,189,86),(78,98,190,87),(79,99,191,88),(80,100,192,89),(81,101,181,90),(82,102,182,91),(83,103,183,92),(84,104,184,93)]])
60 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3 | 4A | ··· | 4L | 4M | ··· | 4T | 6A | ··· | 6G | 12A | ··· | 12X |
| order | 1 | 2 | ··· | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | + | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | Q8 | D6 | C4○D4 | Dic6 | C4×S3 | D12 | C3⋊D4 | C4○D12 |
| kernel | C12⋊4(C4⋊C4) | C6.C42 | C2×Dic3⋊C4 | C2×C4⋊Dic3 | C2×C4×C12 | C4⋊Dic3 | C2×C42 | C2×C12 | C2×C12 | C22×C4 | C2×C6 | C2×C4 | C2×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 2 | 2 | 2 | 1 | 8 | 1 | 4 | 4 | 3 | 4 | 8 | 4 | 4 | 4 | 8 |
Matrix representation of C12⋊4(C4⋊C4) ►in GL5(𝔽13)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 0 | 10 | 6 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 5 | 10 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 3 | 7 |
| 0 | 0 | 0 | 10 | 10 |
| 5 | 0 | 0 | 0 | 0 |
| 0 | 12 | 11 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 8 |
G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,10,0,0,0,3,6],[1,0,0,0,0,0,5,0,0,0,0,10,8,0,0,0,0,0,3,10,0,0,0,7,10],[5,0,0,0,0,0,12,1,0,0,0,11,1,0,0,0,0,0,8,0,0,0,0,0,8] >;
C12⋊4(C4⋊C4) in GAP, Magma, Sage, TeX
C_{12}\rtimes_4(C_4\rtimes C_4) % in TeX
G:=Group("C12:4(C4:C4)"); // GroupNames label
G:=SmallGroup(192,487);
// by ID
G=gap.SmallGroup(192,487);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,120,758,58,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^4=c^4=1,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations