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## G = D12.4C8order 192 = 26·3

### The non-split extension by D12 of C8 acting through Inn(D12)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — D12.4C8
 Chief series C1 — C3 — C6 — C12 — C24 — S3×C8 — C8○D12 — D12.4C8
 Lower central C3 — C6 — D12.4C8
 Upper central C1 — C16 — C2×C16

Generators and relations for D12.4C8
G = < a,b,c | a12=b2=1, c8=a6, bab=a-1, ac=ca, bc=cb >

Subgroups: 152 in 84 conjugacy classes, 51 normal (31 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], S3 [×2], C6, C6, C8 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4 [×3], Q8, Dic3 [×2], C12 [×2], D6 [×2], C2×C6, C16 [×2], C16 [×2], C2×C8, C2×C8 [×2], M4(2) [×3], C4○D4, C3⋊C8 [×2], C24 [×2], Dic6, C4×S3 [×2], D12, C3⋊D4 [×2], C2×C12, C2×C16, C2×C16 [×2], M5(2) [×3], C8○D4, C3⋊C16 [×2], C48 [×2], S3×C8 [×2], C8⋊S3 [×2], C4.Dic3, C2×C24, C4○D12, D4○C16, S3×C16 [×2], D6.C8 [×2], C12.C8, C2×C48, C8○D12, D12.4C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C8 [×4], C2×C4 [×6], C23, D6 [×3], C2×C8 [×6], C22×C4, C4×S3 [×2], C22×S3, C22×C8, S3×C8 [×2], S3×C2×C4, D4○C16, S3×C2×C8, D12.4C8

Smallest permutation representation of D12.4C8
On 96 points
Generators in S96
```(1 28 38 53 85 79 9 20 46 61 93 71)(2 29 39 54 86 80 10 21 47 62 94 72)(3 30 40 55 87 65 11 22 48 63 95 73)(4 31 41 56 88 66 12 23 33 64 96 74)(5 32 42 57 89 67 13 24 34 49 81 75)(6 17 43 58 90 68 14 25 35 50 82 76)(7 18 44 59 91 69 15 26 36 51 83 77)(8 19 45 60 92 70 16 27 37 52 84 78)
(1 38)(2 39)(3 40)(4 41)(5 42)(6 43)(7 44)(8 45)(9 46)(10 47)(11 48)(12 33)(13 34)(14 35)(15 36)(16 37)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(55 73)(56 74)(57 75)(58 76)(59 77)(60 78)(61 79)(62 80)(63 65)(64 66)(81 89)(82 90)(83 91)(84 92)(85 93)(86 94)(87 95)(88 96)
(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)```

`G:=sub<Sym(96)| (1,28,38,53,85,79,9,20,46,61,93,71)(2,29,39,54,86,80,10,21,47,62,94,72)(3,30,40,55,87,65,11,22,48,63,95,73)(4,31,41,56,88,66,12,23,33,64,96,74)(5,32,42,57,89,67,13,24,34,49,81,75)(6,17,43,58,90,68,14,25,35,50,82,76)(7,18,44,59,91,69,15,26,36,51,83,77)(8,19,45,60,92,70,16,27,37,52,84,78), (1,38)(2,39)(3,40)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,48)(12,33)(13,34)(14,35)(15,36)(16,37)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78)(61,79)(62,80)(63,65)(64,66)(81,89)(82,90)(83,91)(84,92)(85,93)(86,94)(87,95)(88,96), (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)>;`

`G:=Group( (1,28,38,53,85,79,9,20,46,61,93,71)(2,29,39,54,86,80,10,21,47,62,94,72)(3,30,40,55,87,65,11,22,48,63,95,73)(4,31,41,56,88,66,12,23,33,64,96,74)(5,32,42,57,89,67,13,24,34,49,81,75)(6,17,43,58,90,68,14,25,35,50,82,76)(7,18,44,59,91,69,15,26,36,51,83,77)(8,19,45,60,92,70,16,27,37,52,84,78), (1,38)(2,39)(3,40)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,48)(12,33)(13,34)(14,35)(15,36)(16,37)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78)(61,79)(62,80)(63,65)(64,66)(81,89)(82,90)(83,91)(84,92)(85,93)(86,94)(87,95)(88,96), (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) );`

`G=PermutationGroup([(1,28,38,53,85,79,9,20,46,61,93,71),(2,29,39,54,86,80,10,21,47,62,94,72),(3,30,40,55,87,65,11,22,48,63,95,73),(4,31,41,56,88,66,12,23,33,64,96,74),(5,32,42,57,89,67,13,24,34,49,81,75),(6,17,43,58,90,68,14,25,35,50,82,76),(7,18,44,59,91,69,15,26,36,51,83,77),(8,19,45,60,92,70,16,27,37,52,84,78)], [(1,38),(2,39),(3,40),(4,41),(5,42),(6,43),(7,44),(8,45),(9,46),(10,47),(11,48),(12,33),(13,34),(14,35),(15,36),(16,37),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72),(55,73),(56,74),(57,75),(58,76),(59,77),(60,78),(61,79),(62,80),(63,65),(64,66),(81,89),(82,90),(83,91),(84,92),(85,93),(86,94),(87,95),(88,96)], [(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)])`

72 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 6A 6B 6C 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 12A 12B 12C 12D 16A ··· 16H 16I 16J 16K 16L 16M ··· 16T 24A ··· 24H 48A ··· 48P order 1 2 2 2 2 3 4 4 4 4 4 6 6 6 8 8 8 8 8 8 8 8 8 8 12 12 12 12 16 ··· 16 16 16 16 16 16 ··· 16 24 ··· 24 48 ··· 48 size 1 1 2 6 6 2 1 1 2 6 6 2 2 2 1 1 1 1 2 2 6 6 6 6 2 2 2 2 1 ··· 1 2 2 2 2 6 ··· 6 2 ··· 2 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C8 C8 C8 S3 D6 D6 C4×S3 C4×S3 S3×C8 S3×C8 D4○C16 D12.4C8 kernel D12.4C8 S3×C16 D6.C8 C12.C8 C2×C48 C8○D12 C8⋊S3 C4.Dic3 C4○D12 Dic6 D12 C3⋊D4 C2×C16 C16 C2×C8 C8 C2×C4 C4 C22 C3 C1 # reps 1 2 2 1 1 1 4 2 2 4 4 8 1 2 1 2 2 4 4 8 16

Matrix representation of D12.4C8 in GL2(𝔽97) generated by

 68 29 68 39
,
 96 0 1 1
,
 18 0 0 18
`G:=sub<GL(2,GF(97))| [68,68,29,39],[96,1,0,1],[18,0,0,18] >;`

D12.4C8 in GAP, Magma, Sage, TeX

`D_{12}._4C_8`
`% in TeX`

`G:=Group("D12.4C8");`
`// GroupNames label`

`G:=SmallGroup(192,460);`
`// by ID`

`G=gap.SmallGroup(192,460);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,58,80,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=b^2=1,c^8=a^6,b*a*b=a^-1,a*c=c*a,b*c=c*b>;`
`// generators/relations`

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