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G = C12.55D4order 96 = 25·3

12nd non-split extension by C12 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C12.55D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C3⋊C8 — C12.55D4
 Lower central C3 — C6 — C12.55D4
 Upper central C1 — C2×C4 — C22×C4

Generators and relations for C12.55D4
G = < a,b,c | a12=1, b4=a6, c2=a9, bab-1=cac-1=a5, cbc-1=a3b3 >

Smallest permutation representation of C12.55D4
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 33 42 22 7 27 48 16)(2 26 43 15 8 32 37 21)(3 31 44 20 9 25 38 14)(4 36 45 13 10 30 39 19)(5 29 46 18 11 35 40 24)(6 34 47 23 12 28 41 17)
(1 13 10 22 7 19 4 16)(2 18 11 15 8 24 5 21)(3 23 12 20 9 17 6 14)(25 38 34 47 31 44 28 41)(26 43 35 40 32 37 29 46)(27 48 36 45 33 42 30 39)```

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + - + - image C1 C2 C2 C4 C4 C8 S3 D4 Dic3 D6 Dic3 M4(2) C3⋊D4 C3⋊C8 C4.Dic3 kernel C12.55D4 C2×C3⋊C8 C22×C12 C2×C12 C22×C6 C2×C6 C22×C4 C12 C2×C4 C2×C4 C23 C6 C4 C22 C2 # reps 1 2 1 2 2 8 1 2 1 1 1 2 4 4 4

Matrix representation of C12.55D4 in GL3(𝔽73) generated by

 27 0 0 0 70 0 0 7 49
,
 10 0 0 0 22 7 0 0 51
,
 63 0 0 0 22 7 0 39 51
`G:=sub<GL(3,GF(73))| [27,0,0,0,70,7,0,0,49],[10,0,0,0,22,0,0,7,51],[63,0,0,0,22,39,0,7,51] >;`

C12.55D4 in GAP, Magma, Sage, TeX

`C_{12}._{55}D_4`
`% in TeX`

`G:=Group("C12.55D4");`
`// GroupNames label`

`G:=SmallGroup(96,37);`
`// by ID`

`G=gap.SmallGroup(96,37);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,121,86,2309]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=1,b^4=a^6,c^2=a^9,b*a*b^-1=c*a*c^-1=a^5,c*b*c^-1=a^3*b^3>;`
`// generators/relations`

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