Copied to
clipboard

G = C12.21C42order 192 = 26·3

14th non-split extension by C12 of C42 acting via C42/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C12.21C42
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C22×C12 — C2×C4.Dic3 — C12.21C42
 Lower central C3 — C12 — C12.21C42
 Upper central C1 — C4 — C2×M4(2)

Generators and relations for C12.21C42
G = < a,b,c | a12=1, b4=c4=a6, bab-1=a5, ac=ca, cbc-1=a9b >

Subgroups: 168 in 86 conjugacy classes, 47 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), C22×C4, C3⋊C8, C24, C2×C12, C2×C12, C22×C6, C2×M4(2), C2×M4(2), C2×C3⋊C8, C4.Dic3, C2×C24, C3×M4(2), C22×C12, C4.10C42, C2×C4.Dic3, C6×M4(2), C12.21C42
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C42, C22⋊C4, C4⋊C4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2.C42, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4.10C42, C6.C42, C12.21C42

Smallest permutation representation of C12.21C42
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 28 4 31 7 34 10 25)(2 33 5 36 8 27 11 30)(3 26 6 29 9 32 12 35)(13 45 22 42 19 39 16 48)(14 38 23 47 20 44 17 41)(15 43 24 40 21 37 18 46)
(1 21 4 24 7 15 10 18)(2 22 5 13 8 16 11 19)(3 23 6 14 9 17 12 20)(25 37 34 46 31 43 28 40)(26 38 35 47 32 44 29 41)(27 39 36 48 33 45 30 42)```

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + - - + + - image C1 C2 C2 C4 C4 S3 D4 Q8 Dic3 D6 C4×S3 D12 C3⋊D4 Dic6 C4.10C42 C12.21C42 kernel C12.21C42 C2×C4.Dic3 C6×M4(2) C2×C3⋊C8 C2×C24 C2×M4(2) C2×C12 C22×C6 C2×C8 C22×C4 C2×C4 C2×C4 C2×C4 C23 C3 C1 # reps 1 2 1 8 4 1 3 1 2 1 4 2 4 2 2 4

Matrix representation of C12.21C42 in GL4(𝔽73) generated by

 3 0 0 0 0 3 0 0 29 0 24 0 29 0 0 24
,
 32 0 71 0 0 0 72 1 24 0 41 0 51 46 41 0
,
 1 71 0 0 60 72 0 0 10 41 0 46 0 41 1 0
`G:=sub<GL(4,GF(73))| [3,0,29,29,0,3,0,0,0,0,24,0,0,0,0,24],[32,0,24,51,0,0,0,46,71,72,41,41,0,1,0,0],[1,60,10,0,71,72,41,41,0,0,0,1,0,0,46,0] >;`

C12.21C42 in GAP, Magma, Sage, TeX

`C_{12}._{21}C_4^2`
`% in TeX`

`G:=Group("C12.21C4^2");`
`// GroupNames label`

`G:=SmallGroup(192,119);`
`// by ID`

`G=gap.SmallGroup(192,119);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,253,64,387,184,1123,136,102,6278]);`
`// Polycyclic`

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

׿
×
𝔽