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## G = D12.34C23order 192 = 26·3

### 15th non-split extension by D12 of C23 acting via C23/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — D12.34C23
 Chief series C1 — C3 — C6 — C12 — D12 — C2×D12 — D4○D12 — D12.34C23
 Lower central C3 — C6 — C12 — D12.34C23
 Upper central C1 — C2 — C4○D4 — 2- 1+4

Generators and relations for D12.34C23
G = < a,b,c,d,e | a12=b2=c2=d2=1, e2=a6, bab=dad=a-1, ac=ca, eae-1=a7, cbc=a6b, dbd=a10b, ebe-1=a3b, cd=dc, ce=ec, ede-1=a9d >

Subgroups: 664 in 258 conjugacy classes, 107 normal (20 characteristic)
C1, C2, C2 [×7], C3, C4, C4 [×3], C4 [×4], C22 [×3], C22 [×7], S3 [×3], C6, C6 [×4], C8 [×4], C2×C4 [×3], C2×C4 [×9], D4, D4 [×3], D4 [×12], Q8, Q8 [×3], Q8 [×4], C23 [×3], Dic3, C12, C12 [×3], C12 [×3], D6 [×6], C2×C6 [×3], C2×C6, C2×C8 [×3], M4(2) [×3], D8 [×3], SD16 [×10], Q16 [×3], C2×D4 [×6], C2×Q8 [×3], C2×Q8, C4○D4, C4○D4 [×3], C4○D4 [×7], C3⋊C8, C3⋊C8 [×3], Dic6, C4×S3 [×3], D12 [×3], D12 [×3], C3⋊D4 [×3], C2×C12 [×3], C2×C12 [×6], C3×D4, C3×D4 [×3], C3×D4 [×3], C3×Q8, C3×Q8 [×3], C3×Q8 [×3], C22×S3 [×3], C8○D4, C2×SD16 [×3], C4○D8 [×3], C8⋊C22 [×3], C8.C22 [×3], 2+ 1+4, 2- 1+4, C2×C3⋊C8 [×3], C4.Dic3 [×3], D4⋊S3 [×3], D4.S3, Q82S3 [×9], C3⋊Q16 [×3], C2×D12 [×3], C4○D12 [×3], S3×D4 [×3], Q83S3, C6×Q8 [×3], C6×Q8, C3×C4○D4, C3×C4○D4 [×3], C3×C4○D4 [×3], D4○SD16, C2×Q82S3 [×3], Q8.11D6 [×3], D4.Dic3, D4⋊D6 [×3], Q8.13D6 [×3], D4○D12, C3×2- 1+4, D12.34C23
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C3⋊D4 [×4], C22×S3 [×7], C22×D4, C2×C3⋊D4 [×6], S3×C23, D4○SD16, C22×C3⋊D4, D12.34C23

Smallest permutation representation of D12.34C23
On 48 points
Generators in S48
```(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)
(1 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 21)(14 20)(15 19)(16 18)(22 24)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(37 45)(38 44)(39 43)(40 42)(46 48)
(1 34)(2 35)(3 36)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 31)(11 32)(12 33)(13 46)(14 47)(15 48)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)
(1 31)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 36)(9 35)(10 34)(11 33)(12 32)(13 46)(14 45)(15 44)(16 43)(17 42)(18 41)(19 40)(20 39)(21 38)(22 37)(23 48)(24 47)
(1 13 7 19)(2 20 8 14)(3 15 9 21)(4 22 10 16)(5 17 11 23)(6 24 12 18)(25 43 31 37)(26 38 32 44)(27 45 33 39)(28 40 34 46)(29 47 35 41)(30 42 36 48)```

`G:=sub<Sym(48)| (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,21)(14,20)(15,19)(16,18)(22,24)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(37,45)(38,44)(39,43)(40,42)(46,48), (1,34)(2,35)(3,36)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,46)(14,47)(15,48)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45), (1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,36)(9,35)(10,34)(11,33)(12,32)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,48)(24,47), (1,13,7,19)(2,20,8,14)(3,15,9,21)(4,22,10,16)(5,17,11,23)(6,24,12,18)(25,43,31,37)(26,38,32,44)(27,45,33,39)(28,40,34,46)(29,47,35,41)(30,42,36,48)>;`

`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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,21)(14,20)(15,19)(16,18)(22,24)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(37,45)(38,44)(39,43)(40,42)(46,48), (1,34)(2,35)(3,36)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,46)(14,47)(15,48)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45), (1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,36)(9,35)(10,34)(11,33)(12,32)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,48)(24,47), (1,13,7,19)(2,20,8,14)(3,15,9,21)(4,22,10,16)(5,17,11,23)(6,24,12,18)(25,43,31,37)(26,38,32,44)(27,45,33,39)(28,40,34,46)(29,47,35,41)(30,42,36,48) );`

`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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,21),(14,20),(15,19),(16,18),(22,24),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(37,45),(38,44),(39,43),(40,42),(46,48)], [(1,34),(2,35),(3,36),(4,25),(5,26),(6,27),(7,28),(8,29),(9,30),(10,31),(11,32),(12,33),(13,46),(14,47),(15,48),(16,37),(17,38),(18,39),(19,40),(20,41),(21,42),(22,43),(23,44),(24,45)], [(1,31),(2,30),(3,29),(4,28),(5,27),(6,26),(7,25),(8,36),(9,35),(10,34),(11,33),(12,32),(13,46),(14,45),(15,44),(16,43),(17,42),(18,41),(19,40),(20,39),(21,38),(22,37),(23,48),(24,47)], [(1,13,7,19),(2,20,8,14),(3,15,9,21),(4,22,10,16),(5,17,11,23),(6,24,12,18),(25,43,31,37),(26,38,32,44),(27,45,33,39),(28,40,34,46),(29,47,35,41),(30,42,36,48)])`

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 8 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 C3⋊D4 C3⋊D4 D4○SD16 D12.34C23 kernel D12.34C23 C2×Q8⋊2S3 Q8.11D6 D4.Dic3 D4⋊D6 Q8.13D6 D4○D12 C3×2- 1+4 2- 1+4 C3×D4 C3×Q8 C2×Q8 C4○D4 D4 Q8 C3 C1 # reps 1 3 3 1 3 3 1 1 1 3 1 3 4 6 2 2 1

Matrix representation of D12.34C23 in GL6(𝔽73)

 1 1 0 0 0 0 72 0 0 0 0 0 0 0 0 1 72 0 0 0 72 0 72 1 0 0 0 0 72 1 0 0 0 0 71 1
,
 72 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 72 0 0 1 0 0 72 0 0 0 0 1 72 0 0 0 0 0 72
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 6 6 67 67 0 0 6 6 6 61 0 0 12 0 0 67 0 0 12 12 0 61
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 67 67 67 12 0 0 67 67 6 6 0 0 0 61 0 6 0 0 61 61 0 12
,
 43 13 0 0 0 0 60 30 0 0 0 0 0 0 0 1 1 72 0 0 0 1 0 72 0 0 72 1 0 0 0 0 0 2 0 72

`G:=sub<GL(6,GF(73))| [1,72,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,72,72,72,71,0,0,0,1,1,1],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,72,72,72,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,6,12,12,0,0,6,6,0,12,0,0,67,6,0,0,0,0,67,61,67,61],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,67,67,0,61,0,0,67,67,61,61,0,0,67,6,0,0,0,0,12,6,6,12],[43,60,0,0,0,0,13,30,0,0,0,0,0,0,0,0,72,0,0,0,1,1,1,2,0,0,1,0,0,0,0,0,72,72,0,72] >;`

D12.34C23 in GAP, Magma, Sage, TeX

`D_{12}._{34}C_2^3`
`% in TeX`

`G:=Group("D12.34C2^3");`
`// GroupNames label`

`G:=SmallGroup(192,1396);`
`// by ID`

`G=gap.SmallGroup(192,1396);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,675,136,1684,235,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^12=b^2=c^2=d^2=1,e^2=a^6,b*a*b=d*a*d=a^-1,a*c=c*a,e*a*e^-1=a^7,c*b*c=a^6*b,d*b*d=a^10*b,e*b*e^-1=a^3*b,c*d=d*c,c*e=e*c,e*d*e^-1=a^9*d>;`
`// generators/relations`

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