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## G = C12.28D12order 288 = 25·32

### 28th non-split extension by C12 of D12 acting via D12/C6=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C12.28D12
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — C6.D12 — C12.28D12
 Lower central C32 — C62 — C12.28D12
 Upper central C1 — C22 — C2×C4

Generators and relations for C12.28D12
G = < a,b,c | a12=b12=c2=1, bab-1=a5, cac=a-1, cbc=a6b-1 >

Subgroups: 866 in 179 conjugacy classes, 52 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], S3 [×8], C6 [×2], C6 [×4], C6 [×3], C2×C4, C2×C4 [×4], D4 [×2], Q8 [×2], C23 [×2], C32, Dic3 [×4], C12 [×4], C12 [×6], D6 [×20], C2×C6 [×2], C2×C6, C42, C22⋊C4 [×4], C2×D4, C2×Q8, C3⋊S3 [×2], C3×C6, C3×C6 [×2], Dic6 [×2], D12 [×8], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×5], C3×Q8 [×2], C22×S3 [×6], C4.4D4, C3×Dic3 [×4], C3×C12 [×2], C2×C3⋊S3 [×6], C62, C4×Dic3, D6⋊C4 [×8], C4×C12, C2×Dic6, C2×D12 [×3], C6×Q8, C3×Dic6 [×2], C6×Dic3 [×4], C12⋊S3 [×2], C6×C12, C22×C3⋊S3 [×2], C427S3, C12.23D4, C6.D12 [×4], Dic3×C12, C6×Dic6, C2×C12⋊S3, C12.28D12
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C4○D4 [×2], D12 [×2], C3⋊D4 [×2], C22×S3 [×2], C4.4D4, S32, C2×D12, C4○D12 [×2], Q83S3 [×2], C2×C3⋊D4, C3⋊D12 [×2], C2×S32, C427S3, C12.23D4, D6.6D6 [×2], C2×C3⋊D12, C12.28D12

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 6G 6H 6I 12A 12B 12C 12D 12E ··· 12J 12K ··· 12R 12S 12T 12U 12V order 1 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 6 6 12 12 12 12 12 ··· 12 12 ··· 12 12 12 12 12 size 1 1 1 1 36 36 2 2 4 2 2 6 6 6 6 12 12 2 ··· 2 4 4 4 2 2 2 2 4 ··· 4 6 ··· 6 12 12 12 12

48 irreducible representations

 dim 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 S3 S3 D4 D6 D6 C4○D4 D12 C3⋊D4 C4○D12 S32 Q8⋊3S3 C3⋊D12 C2×S32 D6.6D6 kernel C12.28D12 C6.D12 Dic3×C12 C6×Dic6 C2×C12⋊S3 C4×Dic3 C2×Dic6 C3×C12 C2×Dic3 C2×C12 C3×C6 C12 C12 C6 C2×C4 C6 C4 C22 C2 # reps 1 4 1 1 1 1 1 2 4 2 4 4 4 8 1 2 2 1 4

Matrix representation of C12.28D12 in GL8(𝔽13)

 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 1 0
,
 5 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 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 1 0 0 0 0 0 0 0 12 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 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 1 0 0 0 0 0 0 0 12 12

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

C12.28D12 in GAP, Magma, Sage, TeX

`C_{12}._{28}D_{12}`
`% in TeX`

`G:=Group("C12.28D12");`
`// GroupNames label`

`G:=SmallGroup(288,512);`
`// by ID`

`G=gap.SmallGroup(288,512);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,141,176,422,100,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=b^12=c^2=1,b*a*b^-1=a^5,c*a*c=a^-1,c*b*c=a^6*b^-1>;`
`// generators/relations`

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