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## G = C12.80D12order 288 = 25·32

### 11st non-split extension by C12 of D12 acting via D12/D6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C12.80D12
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C6×C12 — C3×C4.Dic3 — C12.80D12
 Lower central C32 — C3×C6 — C3×C12 — C12.80D12
 Upper central C1 — C4 — C2×C4

Generators and relations for C12.80D12
G = < a,b,c | a12=b12=c2=1, bab-1=cac=a5, cbc=a3b-1 >

Subgroups: 474 in 108 conjugacy classes, 32 normal (all characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C4 [×3], C22, C22, S3 [×4], C6 [×2], C6 [×4], C8, C2×C4, C2×C4 [×2], D4 [×2], Q8, C32, Dic3 [×6], C12 [×4], C12 [×4], D6 [×4], C2×C6 [×2], C2×C6, C42, M4(2), C4○D4, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, Dic6 [×3], C4×S3 [×4], D12 [×3], C2×Dic3, C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×2], C4≀C2, C3×Dic3 [×2], C3⋊Dic3, C3×C12 [×2], C2×C3⋊S3, C62, C4.Dic3, C4×Dic3, C4×C12, C3×M4(2), C4○D12 [×3], C3×C3⋊C8, C6×Dic3, C324Q8, C4×C3⋊S3, C12⋊S3, C327D4, C6×C12, C424S3, D12⋊C4, C3×C4.Dic3, Dic3×C12, C12.59D6, C12.80D12
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, D4 [×2], D6 [×2], C22⋊C4, C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C4≀C2, S32, D6⋊C4 [×2], C6.D6, C3⋊D12 [×2], C424S3, D12⋊C4, C6.D12, C12.80D12

Smallest permutation representation of C12.80D12
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 32 46 19 9 36 42 23 5 28 38 15)(2 25 47 24 10 29 43 16 6 33 39 20)(3 30 48 17 11 34 44 21 7 26 40 13)(4 35 37 22 12 27 45 14 8 31 41 18)
(1 37)(2 42)(3 47)(4 40)(5 45)(6 38)(7 43)(8 48)(9 41)(10 46)(11 39)(12 44)(13 21)(15 19)(16 24)(18 22)(25 27)(26 32)(28 30)(29 35)(31 33)(34 36)```

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 6E 6F 6G 6H 8A 8B 12A ··· 12F 12G ··· 12K 12L ··· 12S 24A 24B 24C 24D order 1 2 2 2 3 3 3 4 4 4 4 4 4 4 4 6 6 6 6 6 6 6 6 8 8 12 ··· 12 12 ··· 12 12 ··· 12 24 24 24 24 size 1 1 2 36 2 2 4 1 1 2 6 6 6 6 36 2 2 2 2 4 4 4 4 12 12 2 ··· 2 4 ··· 4 6 ··· 6 12 12 12 12

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C4 C4 S3 S3 D4 D4 D6 C4×S3 D12 C3⋊D4 D12 C3⋊D4 C4≀C2 C42⋊4S3 S32 C6.D6 C3⋊D12 C3⋊D12 D12⋊C4 C12.80D12 kernel C12.80D12 C3×C4.Dic3 Dic3×C12 C12.59D6 C32⋊4Q8 C12⋊S3 C4.Dic3 C4×Dic3 C3×C12 C62 C2×C12 C12 C12 C12 C2×C6 C2×C6 C32 C3 C2×C4 C4 C4 C22 C3 C1 # reps 1 1 1 1 2 2 1 1 1 1 2 4 2 2 2 2 4 8 1 1 1 1 2 4

Matrix representation of C12.80D12 in GL6(𝔽73)

 46 0 0 0 0 0 0 46 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 1 72
,
 0 45 0 0 0 0 14 45 0 0 0 0 0 0 1 72 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 72 2 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 72 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0

`G:=sub<GL(6,GF(73))| [46,0,0,0,0,0,0,46,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[0,14,0,0,0,0,45,45,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[72,0,0,0,0,0,2,1,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;`

C12.80D12 in GAP, Magma, Sage, TeX

`C_{12}._{80}D_{12}`
`% in TeX`

`G:=Group("C12.80D12");`
`// GroupNames label`

`G:=SmallGroup(288,218);`
`// by ID`

`G=gap.SmallGroup(288,218);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,92,100,675,80,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=b^12=c^2=1,b*a*b^-1=c*a*c=a^5,c*b*c=a^3*b^-1>;`
`// generators/relations`

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