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## G = C12⋊4D12order 288 = 25·32

### 1st semidirect product of C12 and D12 acting via D12/C12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C12⋊4D12
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C22×C3⋊S3 — C2×C12⋊S3 — C12⋊4D12
 Lower central C32 — C62 — C12⋊4D12
 Upper central C1 — C22 — C42

Generators and relations for C124D12
G = < a,b,c | a12=b12=c2=1, ab=ba, cac=a-1, cbc=b-1 >

Subgroups: 1740 in 324 conjugacy classes, 101 normal (7 characteristic)
C1, C2 [×3], C2 [×4], C3 [×4], C4 [×6], C22, C22 [×12], S3 [×16], C6 [×12], C2×C4 [×3], D4 [×12], C23 [×4], C32, C12 [×24], D6 [×48], C2×C6 [×4], C42, C2×D4 [×6], C3⋊S3 [×4], C3×C6 [×3], D12 [×48], C2×C12 [×12], C22×S3 [×16], C41D4, C3×C12 [×6], C2×C3⋊S3 [×12], C62, C4×C12 [×4], C2×D12 [×24], C12⋊S3 [×12], C6×C12 [×3], C22×C3⋊S3 [×4], C4⋊D12 [×4], C122, C2×C12⋊S3 [×6], C124D12
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×6], C23, D6 [×12], C2×D4 [×3], C3⋊S3, D12 [×24], C22×S3 [×4], C41D4, C2×C3⋊S3 [×3], C2×D12 [×12], C12⋊S3 [×6], C22×C3⋊S3, C4⋊D12 [×4], C2×C12⋊S3 [×3], C124D12

Smallest permutation representation of C124D12
On 144 points
Generators in S144
```(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)
(1 120 74 28 55 63 108 123 94 16 45 142)(2 109 75 29 56 64 97 124 95 17 46 143)(3 110 76 30 57 65 98 125 96 18 47 144)(4 111 77 31 58 66 99 126 85 19 48 133)(5 112 78 32 59 67 100 127 86 20 37 134)(6 113 79 33 60 68 101 128 87 21 38 135)(7 114 80 34 49 69 102 129 88 22 39 136)(8 115 81 35 50 70 103 130 89 23 40 137)(9 116 82 36 51 71 104 131 90 24 41 138)(10 117 83 25 52 72 105 132 91 13 42 139)(11 118 84 26 53 61 106 121 92 14 43 140)(12 119 73 27 54 62 107 122 93 15 44 141)
(1 133)(2 144)(3 143)(4 142)(5 141)(6 140)(7 139)(8 138)(9 137)(10 136)(11 135)(12 134)(13 80)(14 79)(15 78)(16 77)(17 76)(18 75)(19 74)(20 73)(21 84)(22 83)(23 82)(24 81)(25 88)(26 87)(27 86)(28 85)(29 96)(30 95)(31 94)(32 93)(33 92)(34 91)(35 90)(36 89)(37 119)(38 118)(39 117)(40 116)(41 115)(42 114)(43 113)(44 112)(45 111)(46 110)(47 109)(48 120)(49 132)(50 131)(51 130)(52 129)(53 128)(54 127)(55 126)(56 125)(57 124)(58 123)(59 122)(60 121)(61 101)(62 100)(63 99)(64 98)(65 97)(66 108)(67 107)(68 106)(69 105)(70 104)(71 103)(72 102)```

`G:=sub<Sym(144)| (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), (1,120,74,28,55,63,108,123,94,16,45,142)(2,109,75,29,56,64,97,124,95,17,46,143)(3,110,76,30,57,65,98,125,96,18,47,144)(4,111,77,31,58,66,99,126,85,19,48,133)(5,112,78,32,59,67,100,127,86,20,37,134)(6,113,79,33,60,68,101,128,87,21,38,135)(7,114,80,34,49,69,102,129,88,22,39,136)(8,115,81,35,50,70,103,130,89,23,40,137)(9,116,82,36,51,71,104,131,90,24,41,138)(10,117,83,25,52,72,105,132,91,13,42,139)(11,118,84,26,53,61,106,121,92,14,43,140)(12,119,73,27,54,62,107,122,93,15,44,141), (1,133)(2,144)(3,143)(4,142)(5,141)(6,140)(7,139)(8,138)(9,137)(10,136)(11,135)(12,134)(13,80)(14,79)(15,78)(16,77)(17,76)(18,75)(19,74)(20,73)(21,84)(22,83)(23,82)(24,81)(25,88)(26,87)(27,86)(28,85)(29,96)(30,95)(31,94)(32,93)(33,92)(34,91)(35,90)(36,89)(37,119)(38,118)(39,117)(40,116)(41,115)(42,114)(43,113)(44,112)(45,111)(46,110)(47,109)(48,120)(49,132)(50,131)(51,130)(52,129)(53,128)(54,127)(55,126)(56,125)(57,124)(58,123)(59,122)(60,121)(61,101)(62,100)(63,99)(64,98)(65,97)(66,108)(67,107)(68,106)(69,105)(70,104)(71,103)(72,102)>;`

`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), (1,120,74,28,55,63,108,123,94,16,45,142)(2,109,75,29,56,64,97,124,95,17,46,143)(3,110,76,30,57,65,98,125,96,18,47,144)(4,111,77,31,58,66,99,126,85,19,48,133)(5,112,78,32,59,67,100,127,86,20,37,134)(6,113,79,33,60,68,101,128,87,21,38,135)(7,114,80,34,49,69,102,129,88,22,39,136)(8,115,81,35,50,70,103,130,89,23,40,137)(9,116,82,36,51,71,104,131,90,24,41,138)(10,117,83,25,52,72,105,132,91,13,42,139)(11,118,84,26,53,61,106,121,92,14,43,140)(12,119,73,27,54,62,107,122,93,15,44,141), (1,133)(2,144)(3,143)(4,142)(5,141)(6,140)(7,139)(8,138)(9,137)(10,136)(11,135)(12,134)(13,80)(14,79)(15,78)(16,77)(17,76)(18,75)(19,74)(20,73)(21,84)(22,83)(23,82)(24,81)(25,88)(26,87)(27,86)(28,85)(29,96)(30,95)(31,94)(32,93)(33,92)(34,91)(35,90)(36,89)(37,119)(38,118)(39,117)(40,116)(41,115)(42,114)(43,113)(44,112)(45,111)(46,110)(47,109)(48,120)(49,132)(50,131)(51,130)(52,129)(53,128)(54,127)(55,126)(56,125)(57,124)(58,123)(59,122)(60,121)(61,101)(62,100)(63,99)(64,98)(65,97)(66,108)(67,107)(68,106)(69,105)(70,104)(71,103)(72,102) );`

`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)], [(1,120,74,28,55,63,108,123,94,16,45,142),(2,109,75,29,56,64,97,124,95,17,46,143),(3,110,76,30,57,65,98,125,96,18,47,144),(4,111,77,31,58,66,99,126,85,19,48,133),(5,112,78,32,59,67,100,127,86,20,37,134),(6,113,79,33,60,68,101,128,87,21,38,135),(7,114,80,34,49,69,102,129,88,22,39,136),(8,115,81,35,50,70,103,130,89,23,40,137),(9,116,82,36,51,71,104,131,90,24,41,138),(10,117,83,25,52,72,105,132,91,13,42,139),(11,118,84,26,53,61,106,121,92,14,43,140),(12,119,73,27,54,62,107,122,93,15,44,141)], [(1,133),(2,144),(3,143),(4,142),(5,141),(6,140),(7,139),(8,138),(9,137),(10,136),(11,135),(12,134),(13,80),(14,79),(15,78),(16,77),(17,76),(18,75),(19,74),(20,73),(21,84),(22,83),(23,82),(24,81),(25,88),(26,87),(27,86),(28,85),(29,96),(30,95),(31,94),(32,93),(33,92),(34,91),(35,90),(36,89),(37,119),(38,118),(39,117),(40,116),(41,115),(42,114),(43,113),(44,112),(45,111),(46,110),(47,109),(48,120),(49,132),(50,131),(51,130),(52,129),(53,128),(54,127),(55,126),(56,125),(57,124),(58,123),(59,122),(60,121),(61,101),(62,100),(63,99),(64,98),(65,97),(66,108),(67,107),(68,106),(69,105),(70,104),(71,103),(72,102)])`

78 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 4A ··· 4F 6A ··· 6L 12A ··· 12AV order 1 2 2 2 2 2 2 2 3 3 3 3 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 1 1 36 36 36 36 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

78 irreducible representations

 dim 1 1 1 2 2 2 2 type + + + + + + + image C1 C2 C2 S3 D4 D6 D12 kernel C12⋊4D12 C122 C2×C12⋊S3 C4×C12 C3×C12 C2×C12 C12 # reps 1 1 6 4 6 12 48

Matrix representation of C124D12 in GL4(𝔽13) generated by

 0 12 0 0 1 1 0 0 0 0 10 6 0 0 7 3
,
 3 6 0 0 7 10 0 0 0 0 6 10 0 0 3 3
,
 10 7 0 0 10 3 0 0 0 0 12 1 0 0 0 1
`G:=sub<GL(4,GF(13))| [0,1,0,0,12,1,0,0,0,0,10,7,0,0,6,3],[3,7,0,0,6,10,0,0,0,0,6,3,0,0,10,3],[10,10,0,0,7,3,0,0,0,0,12,0,0,0,1,1] >;`

C124D12 in GAP, Magma, Sage, TeX

`C_{12}\rtimes_4D_{12}`
`% in TeX`

`G:=Group("C12:4D12");`
`// GroupNames label`

`G:=SmallGroup(288,731);`
`// by ID`

`G=gap.SmallGroup(288,731);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,254,58,2693,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=b^12=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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