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## G = D12.23D4order 192 = 26·3

### 6th non-split extension by D12 of D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — D12.23D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C4×D12 — D12.23D4
 Lower central C3 — C6 — C2×C12 — D12.23D4
 Upper central C1 — C22 — C42 — C4.4D4

Generators and relations for D12.23D4
G = < a,b,c,d | a12=b2=c4=1, d2=a6, bab=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a3b, dcd-1=a6c-1 >

Subgroups: 384 in 124 conjugacy classes, 41 normal (39 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×2], C4 [×4], C22, C22 [×7], S3 [×2], C6 [×3], C6, C8 [×2], C2×C4 [×3], C2×C4 [×4], D4 [×5], Q8 [×2], C23 [×2], Dic3, C12 [×2], C12 [×3], D6 [×4], C2×C6, C2×C6 [×3], C42, C22⋊C4 [×3], C4⋊C4, C2×C8 [×2], D8 [×2], SD16 [×2], C22×C4, C2×D4, C2×D4, C2×Q8, C3⋊C8 [×2], C4×S3 [×2], D12 [×2], D12, C2×Dic3, C2×C12 [×3], C2×C12, C3×D4 [×2], C3×Q8 [×2], C22×S3, C22×C6, D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C2×C3⋊C8 [×2], C4⋊Dic3, D6⋊C4, D4⋊S3 [×2], Q82S3 [×2], C4×C12, C3×C22⋊C4 [×2], S3×C2×C4, C2×D12, C6×D4, C6×Q8, D4.2D4, C12⋊C8, D4⋊Dic3, Q82Dic3, C4×D12, C2×D4⋊S3, C2×Q82S3, C3×C4.4D4, D12.23D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4, C3⋊D4 [×2], C22×S3, C4⋊D4, C4○D8, C8⋊C22, S3×D4, D42S3, C2×C3⋊D4, D4.2D4, D63D4, D4⋊D6, Q8.13D6, D12.23D4

Smallest permutation representation of D12.23D4
On 96 points
Generators in S96
```(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)
(1 93)(2 92)(3 91)(4 90)(5 89)(6 88)(7 87)(8 86)(9 85)(10 96)(11 95)(12 94)(13 63)(14 62)(15 61)(16 72)(17 71)(18 70)(19 69)(20 68)(21 67)(22 66)(23 65)(24 64)(25 60)(26 59)(27 58)(28 57)(29 56)(30 55)(31 54)(32 53)(33 52)(34 51)(35 50)(36 49)(37 81)(38 80)(39 79)(40 78)(41 77)(42 76)(43 75)(44 74)(45 73)(46 84)(47 83)(48 82)
(1 15 94 62)(2 16 95 63)(3 17 96 64)(4 18 85 65)(5 19 86 66)(6 20 87 67)(7 21 88 68)(8 22 89 69)(9 23 90 70)(10 24 91 71)(11 13 92 72)(12 14 93 61)(25 75 56 39)(26 76 57 40)(27 77 58 41)(28 78 59 42)(29 79 60 43)(30 80 49 44)(31 81 50 45)(32 82 51 46)(33 83 52 47)(34 84 53 48)(35 73 54 37)(36 74 55 38)
(1 79 7 73)(2 74 8 80)(3 81 9 75)(4 76 10 82)(5 83 11 77)(6 78 12 84)(13 33 19 27)(14 28 20 34)(15 35 21 29)(16 30 22 36)(17 25 23 31)(18 32 24 26)(37 94 43 88)(38 89 44 95)(39 96 45 90)(40 91 46 85)(41 86 47 92)(42 93 48 87)(49 69 55 63)(50 64 56 70)(51 71 57 65)(52 66 58 72)(53 61 59 67)(54 68 60 62)```

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

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

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

33 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 6E 8A 8B 8C 8D 12A ··· 12F 12G 12H order 1 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 6 6 6 6 6 8 8 8 8 12 ··· 12 12 12 size 1 1 1 1 8 12 12 2 2 2 2 2 4 8 12 12 2 2 2 8 8 12 12 12 12 4 ··· 4 8 8

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 C4○D4 C3⋊D4 C4○D8 C8⋊C22 S3×D4 D4⋊2S3 D4⋊D6 Q8.13D6 kernel D12.23D4 C12⋊C8 D4⋊Dic3 Q8⋊2Dic3 C4×D12 C2×D4⋊S3 C2×Q8⋊2S3 C3×C4.4D4 C4.4D4 D12 C2×C12 C42 C2×D4 C2×Q8 C12 C2×C4 C6 C6 C4 C4 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 2 1 1 1 2 4 4 1 1 1 2 2

Matrix representation of D12.23D4 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 1 72 0 0 0 0 0 0 1 71 0 0 0 0 1 72
,
 1 0 0 0 0 0 0 72 0 0 0 0 0 0 1 72 0 0 0 0 0 72 0 0 0 0 0 0 72 2 0 0 0 0 0 1
,
 46 0 0 0 0 0 0 27 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 27 0 0 0 0 0 0 27
,
 0 27 0 0 0 0 46 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 12 61 0 0 0 0 6 61

`G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,1,0,0,0,0,71,72],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,0,0,0,72,0,0,0,0,0,2,1],[46,0,0,0,0,0,0,27,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,27,0,0,0,0,0,0,27],[0,46,0,0,0,0,27,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,12,6,0,0,0,0,61,61] >;`

D12.23D4 in GAP, Magma, Sage, TeX

`D_{12}._{23}D_4`
`% in TeX`

`G:=Group("D12.23D4");`
`// GroupNames label`

`G:=SmallGroup(192,616);`
`// by ID`

`G=gap.SmallGroup(192,616);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,344,254,219,1123,297,136,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=c^4=1,d^2=a^6,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=a^6*c^-1>;`
`// generators/relations`

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