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## G = C12⋊3D12order 288 = 25·32

### 3rd semidirect product of C12 and D12 acting via D12/C6=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C12⋊3D12
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C22×C3⋊S3 — C2×C4×C3⋊S3 — C12⋊3D12
 Lower central C32 — C62 — C12⋊3D12
 Upper central C1 — C22 — C4⋊C4

Generators and relations for C123D12
G = < a,b,c | a12=b12=c2=1, bab-1=a7, cac=a-1, cbc=b-1 >

Subgroups: 1388 in 282 conjugacy classes, 79 normal (19 characteristic)
C1, C2 [×3], C2 [×4], C3 [×4], C4 [×2], C4 [×3], C22, C22 [×10], S3 [×16], C6 [×12], C2×C4, C2×C4 [×2], C2×C4 [×3], D4 [×6], C23 [×3], C32, Dic3 [×4], C12 [×8], C12 [×8], D6 [×40], C2×C6 [×4], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], C3⋊S3 [×4], C3×C6 [×3], C4×S3 [×8], D12 [×24], C2×Dic3 [×4], C2×C12 [×12], C22×S3 [×12], C4⋊D4, C3⋊Dic3, C3×C12 [×2], C3×C12 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×8], C62, D6⋊C4 [×8], C3×C4⋊C4 [×4], S3×C2×C4 [×4], C2×D12 [×12], C4×C3⋊S3 [×2], C12⋊S3 [×6], C2×C3⋊Dic3, C6×C12, C6×C12 [×2], C22×C3⋊S3, C22×C3⋊S3 [×2], C12⋊D4 [×4], C6.11D12 [×2], C32×C4⋊C4, C2×C4×C3⋊S3, C2×C12⋊S3, C2×C12⋊S3 [×2], C123D12
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×4], C23, D6 [×12], C2×D4 [×2], C4○D4, C3⋊S3, D12 [×8], C22×S3 [×4], C4⋊D4, C2×C3⋊S3 [×3], C2×D12 [×4], S3×D4 [×4], Q83S3 [×4], C12⋊S3 [×2], C22×C3⋊S3, C12⋊D4 [×4], C2×C12⋊S3, D4×C3⋊S3, C12.26D6, C123D12

Smallest permutation representation of C123D12
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 48 33 67 91 56 119 21 80 106 138 132)(2 43 34 62 92 51 120 16 81 101 139 127)(3 38 35 69 93 58 109 23 82 108 140 122)(4 45 36 64 94 53 110 18 83 103 141 129)(5 40 25 71 95 60 111 13 84 98 142 124)(6 47 26 66 96 55 112 20 73 105 143 131)(7 42 27 61 85 50 113 15 74 100 144 126)(8 37 28 68 86 57 114 22 75 107 133 121)(9 44 29 63 87 52 115 17 76 102 134 128)(10 39 30 70 88 59 116 24 77 97 135 123)(11 46 31 65 89 54 117 19 78 104 136 130)(12 41 32 72 90 49 118 14 79 99 137 125)
(1 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 36)(11 35)(12 34)(13 17)(14 16)(18 24)(19 23)(20 22)(37 47)(38 46)(39 45)(40 44)(41 43)(49 101)(50 100)(51 99)(52 98)(53 97)(54 108)(55 107)(56 106)(57 105)(58 104)(59 103)(60 102)(61 126)(62 125)(63 124)(64 123)(65 122)(66 121)(67 132)(68 131)(69 130)(70 129)(71 128)(72 127)(73 114)(74 113)(75 112)(76 111)(77 110)(78 109)(79 120)(80 119)(81 118)(82 117)(83 116)(84 115)(85 144)(86 143)(87 142)(88 141)(89 140)(90 139)(91 138)(92 137)(93 136)(94 135)(95 134)(96 133)```

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 6A ··· 6L 12A ··· 12X order 1 2 2 2 2 2 2 2 3 3 3 3 4 4 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 1 1 18 18 36 36 2 2 2 2 2 2 4 4 18 18 2 ··· 2 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D4 D6 C4○D4 D12 S3×D4 Q8⋊3S3 kernel C12⋊3D12 C6.11D12 C32×C4⋊C4 C2×C4×C3⋊S3 C2×C12⋊S3 C3×C4⋊C4 C3×C12 C2×C3⋊S3 C2×C12 C3×C6 C12 C6 C6 # reps 1 2 1 1 3 4 2 2 12 2 16 4 4

Matrix representation of C123D12 in GL8(𝔽13)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 11 7 0 0 0 0 0 0 3 2
,
 12 10 0 0 0 0 0 0 5 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 5 1
,
 12 0 0 0 0 0 0 0 5 1 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 8 12

`G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,11,3,0,0,0,0,0,0,7,2],[12,5,0,0,0,0,0,0,10,1,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,12,5,0,0,0,0,0,0,0,1],[12,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,8,0,0,0,0,0,0,0,12] >;`

C123D12 in GAP, Magma, Sage, TeX

`C_{12}\rtimes_3D_{12}`
`% in TeX`

`G:=Group("C12:3D12");`
`// GroupNames label`

`G:=SmallGroup(288,752);`
`// by ID`

`G=gap.SmallGroup(288,752);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,219,58,2693,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=b^12=c^2=1,b*a*b^-1=a^7,c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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