Copied to
clipboard

G = C62.4C8order 288 = 25·32

2nd non-split extension by C62 of C8 acting via C8/C2=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C62.4C8
 Chief series C1 — C32 — C3×C6 — C3×C12 — C32⋊4C8 — C32⋊2C16 — C62.4C8
 Lower central C32 — C3×C6 — C62.4C8
 Upper central C1 — C4 — C2×C4

Generators and relations for C62.4C8
G = < a,b,c | a6=b6=1, c8=b3, ab=ba, cac-1=a-1b, cbc-1=a4b >

Smallest permutation representation of C62.4C8
On 48 points
Generators in S48
```(1 27 47)(2 40 28 10 48 20)(3 33 29)(4 22 34 12 30 42)(5 31 35)(6 44 32 14 36 24)(7 37 17)(8 26 38 16 18 46)(9 19 39)(11 41 21)(13 23 43)(15 45 25)
(1 9)(2 20 48 10 28 40)(3 11)(4 42 30 12 34 22)(5 13)(6 24 36 14 32 44)(7 15)(8 46 18 16 38 26)(17 25)(19 27)(21 29)(23 31)(33 41)(35 43)(37 45)(39 47)
(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)```

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

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

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

36 conjugacy classes

 class 1 2A 2B 3A 3B 4A 4B 4C 6A ··· 6F 8A 8B 8C 8D 8E 8F 12A ··· 12H 16A ··· 16H order 1 2 2 3 3 4 4 4 6 ··· 6 8 8 8 8 8 8 12 ··· 12 16 ··· 16 size 1 1 2 4 4 1 1 2 4 ··· 4 9 9 9 9 18 18 4 ··· 4 18 ··· 18

36 irreducible representations

 dim 1 1 1 1 1 1 1 2 4 4 4 4 4 type + + + + - + - image C1 C2 C2 C4 C4 C8 C8 M5(2) C32⋊C4 C32⋊2C8 C2×C32⋊C4 C32⋊2C8 C62.4C8 kernel C62.4C8 C32⋊2C16 C2×C32⋊4C8 C32⋊4C8 C6×C12 C3×C12 C62 C32 C2×C4 C4 C4 C22 C1 # reps 1 2 1 2 2 4 4 4 2 2 2 2 8

Matrix representation of C62.4C8 in GL4(𝔽5) generated by

 1 0 0 3 0 2 4 0 0 3 4 0 4 0 0 3
,
 4 0 0 0 0 4 1 0 0 2 2 0 0 0 0 4
,
 0 0 4 0 1 0 0 0 0 0 0 2 0 1 0 0
`G:=sub<GL(4,GF(5))| [1,0,0,4,0,2,3,0,0,4,4,0,3,0,0,3],[4,0,0,0,0,4,2,0,0,1,2,0,0,0,0,4],[0,1,0,0,0,0,0,1,4,0,0,0,0,0,2,0] >;`

C62.4C8 in GAP, Magma, Sage, TeX

`C_6^2._4C_8`
`% in TeX`

`G:=Group("C6^2.4C8");`
`// GroupNames label`

`G:=SmallGroup(288,421);`
`// by ID`

`G=gap.SmallGroup(288,421);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,253,58,80,9413,691,12550,2372]);`
`// Polycyclic`

`G:=Group<a,b,c|a^6=b^6=1,c^8=b^3,a*b=b*a,c*a*c^-1=a^-1*b,c*b*c^-1=a^4*b>;`
`// generators/relations`

Export

׿
×
𝔽