Copied to
clipboard

## G = C33⋊4M4(2)  order 432 = 24·33

### 2nd semidirect product of C33 and M4(2) acting via M4(2)/C4=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C33⋊4M4(2)
 Chief series C1 — C3 — C33 — C32×C6 — C3×C3⋊Dic3 — C33⋊4C8 — C33⋊4M4(2)
 Lower central C33 — C32×C6 — C33⋊4M4(2)
 Upper central C1 — C2 — C4

Generators and relations for C334M4(2)
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, dbd-1=ab=ba, ac=ca, dad-1=a-1b, eae=a-1, bc=cb, ebe=b-1, dcd-1=c-1, ce=ec, ede=d5 >

Subgroups: 424 in 84 conjugacy classes, 21 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, C32, C32, Dic3, C12, C12, D6, C2×C6, M4(2), C3×S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C4×S3, C2×C12, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C4.Dic3, C3×C3⋊S3, C32×C6, C322C8, S3×C12, C4×C3⋊S3, C3×C3⋊Dic3, C32×C12, C6×C3⋊S3, C32⋊M4(2), C334C8, C12×C3⋊S3, C334M4(2)
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, M4(2), C2×Dic3, C32⋊C4, C4.Dic3, C2×C32⋊C4, C33⋊C4, C32⋊M4(2), C2×C33⋊C4, C334M4(2)

Smallest permutation representation of C334M4(2)
On 48 points
Generators in S48
```(2 16 38)(4 40 10)(6 12 34)(8 36 14)(18 45 31)(20 25 47)(22 41 27)(24 29 43)
(1 15 37)(2 16 38)(3 39 9)(4 40 10)(5 11 33)(6 12 34)(7 35 13)(8 36 14)(17 44 30)(18 45 31)(19 32 46)(20 25 47)(21 48 26)(22 41 27)(23 28 42)(24 29 43)
(1 37 15)(2 16 38)(3 39 9)(4 10 40)(5 33 11)(6 12 34)(7 35 13)(8 14 36)(17 30 44)(18 45 31)(19 32 46)(20 47 25)(21 26 48)(22 41 27)(23 28 42)(24 43 29)
(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 23)(2 20)(3 17)(4 22)(5 19)(6 24)(7 21)(8 18)(9 44)(10 41)(11 46)(12 43)(13 48)(14 45)(15 42)(16 47)(25 38)(26 35)(27 40)(28 37)(29 34)(30 39)(31 36)(32 33)```

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

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

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

42 conjugacy classes

 class 1 2A 2B 3A 3B ··· 3G 4A 4B 4C 6A 6B ··· 6G 6H 6I 8A 8B 8C 8D 12A 12B 12C ··· 12N 12O 12P order 1 2 2 3 3 ··· 3 4 4 4 6 6 ··· 6 6 6 8 8 8 8 12 12 12 ··· 12 12 12 size 1 1 18 2 4 ··· 4 2 9 9 2 4 ··· 4 18 18 54 54 54 54 2 2 4 ··· 4 18 18

42 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + - - + + image C1 C2 C2 C4 C4 S3 D6 Dic3 Dic3 M4(2) C4.Dic3 C32⋊C4 C2×C32⋊C4 C33⋊C4 C32⋊M4(2) C2×C33⋊C4 C33⋊4M4(2) kernel C33⋊4M4(2) C33⋊4C8 C12×C3⋊S3 C32×C12 C6×C3⋊S3 C4×C3⋊S3 C3⋊Dic3 C3×C12 C2×C3⋊S3 C33 C32 C12 C6 C4 C3 C2 C1 # reps 1 2 1 2 2 1 1 1 1 2 4 2 2 4 4 4 8

Matrix representation of C334M4(2) in GL4(𝔽73) generated by

 1 0 68 54 0 1 59 44 0 0 8 0 0 0 0 64
,
 8 0 0 48 0 64 39 0 0 0 8 0 0 0 0 64
,
 64 0 40 25 0 64 39 42 0 0 8 0 0 0 0 8
,
 11 16 19 69 60 59 11 58 43 0 62 57 0 43 13 14
,
 0 27 25 63 46 0 10 25 0 0 0 46 0 0 27 0
`G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,68,59,8,0,54,44,0,64],[8,0,0,0,0,64,0,0,0,39,8,0,48,0,0,64],[64,0,0,0,0,64,0,0,40,39,8,0,25,42,0,8],[11,60,43,0,16,59,0,43,19,11,62,13,69,58,57,14],[0,46,0,0,27,0,0,0,25,10,0,27,63,25,46,0] >;`

C334M4(2) in GAP, Magma, Sage, TeX

`C_3^3\rtimes_4M_4(2)`
`% in TeX`

`G:=Group("C3^3:4M4(2)");`
`// GroupNames label`

`G:=SmallGroup(432,636);`
`// by ID`

`G=gap.SmallGroup(432,636);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,28,141,64,58,2804,298,2693,1027,14118]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=e^2=1,d*b*d^-1=a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b,e*a*e=a^-1,b*c=c*b,e*b*e=b^-1,d*c*d^-1=c^-1,c*e=e*c,e*d*e=d^5>;`
`// generators/relations`

׿
×
𝔽