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G = C8○D12order 96 = 25·3

Central product of C8 and D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C8○D12
 Chief series C1 — C3 — C6 — C12 — C4×S3 — C4○D12 — C8○D12
 Lower central C3 — C6 — C8○D12
 Upper central C1 — C8 — C2×C8

Generators and relations for C8○D12
G = < a,b,c | a8=c2=1, b6=a4, ab=ba, ac=ca, cbc=a4b5 >

Subgroups: 114 in 62 conjugacy classes, 37 normal (23 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, Dic3, C12, D6, C2×C6, C2×C8, C2×C8, M4(2), C4○D4, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C8○D4, S3×C8, C8⋊S3, C4.Dic3, C2×C24, C4○D12, C8○D12
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4×S3, C22×S3, C8○D4, S3×C2×C4, C8○D12

Smallest permutation representation of C8○D12
On 48 points
Generators in S48
```(1 21 45 36 7 15 39 30)(2 22 46 25 8 16 40 31)(3 23 47 26 9 17 41 32)(4 24 48 27 10 18 42 33)(5 13 37 28 11 19 43 34)(6 14 38 29 12 20 44 35)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 19)(14 18)(15 17)(20 24)(21 23)(26 36)(27 35)(28 34)(29 33)(30 32)(37 43)(38 42)(39 41)(44 48)(45 47)```

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

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

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

36 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 24A ··· 24H 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 24 ··· 24 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 2 ··· 2

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 S3 D6 D6 C4×S3 C4×S3 C8○D4 C8○D12 kernel C8○D12 S3×C8 C8⋊S3 C4.Dic3 C2×C24 C4○D12 Dic6 D12 C3⋊D4 C2×C8 C8 C2×C4 C4 C22 C3 C1 # reps 1 2 2 1 1 1 2 2 4 1 2 1 2 2 4 8

Matrix representation of C8○D12 in GL2(𝔽73) generated by

 63 0 0 63
,
 66 7 66 59
,
 1 1 0 72
`G:=sub<GL(2,GF(73))| [63,0,0,63],[66,66,7,59],[1,0,1,72] >;`

C8○D12 in GAP, Magma, Sage, TeX

`C_8\circ D_{12}`
`% in TeX`

`G:=Group("C8oD12");`
`// GroupNames label`

`G:=SmallGroup(96,108);`
`// by ID`

`G=gap.SmallGroup(96,108);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,50,69,2309]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=c^2=1,b^6=a^4,a*b=b*a,a*c=c*a,c*b*c=a^4*b^5>;`
`// generators/relations`

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