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## G = M5(2)⋊S3order 192 = 26·3

### 5th semidirect product of M5(2) and S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — M5(2)⋊S3
 Chief series C1 — C3 — C6 — C12 — C24 — C2×C24 — C2×D24 — M5(2)⋊S3
 Lower central C3 — C6 — C12 — C24 — M5(2)⋊S3
 Upper central C1 — C2 — C2×C4 — C2×C8 — M5(2)

Generators and relations for M5(2)⋊S3
G = < a,b,c,d | a16=b2=c3=d2=1, bab=a9, ac=ca, dad=a11b, bc=cb, bd=db, dcd=c-1 >

Subgroups: 296 in 62 conjugacy classes, 25 normal (23 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, D4, C23, C12, D6, C2×C6, C16, C2×C8, M4(2), D8, C2×D4, C3⋊C8, C24, D12, C2×C12, C22×S3, C8.C4, M5(2), C2×D8, C48, D24, D24, C4.Dic3, C2×C24, C2×D12, M5(2)⋊C2, C24.C4, C3×M5(2), C2×D24, M5(2)⋊S3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, D8, SD16, C4×S3, D12, C3⋊D4, D4⋊C4, C24⋊C2, D24, D6⋊C4, M5(2)⋊C2, C2.D24, M5(2)⋊S3

Smallest permutation representation of M5(2)⋊S3
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)
(2 10)(4 12)(6 14)(8 16)(17 25)(19 27)(21 29)(23 31)(33 41)(35 43)(37 45)(39 47)
(1 36 30)(2 37 31)(3 38 32)(4 39 17)(5 40 18)(6 41 19)(7 42 20)(8 43 21)(9 44 22)(10 45 23)(11 46 24)(12 47 25)(13 48 26)(14 33 27)(15 34 28)(16 35 29)
(2 4)(3 15)(5 13)(6 16)(7 11)(8 14)(10 12)(17 37)(18 48)(19 35)(20 46)(21 33)(22 44)(23 47)(24 42)(25 45)(26 40)(27 43)(28 38)(29 41)(30 36)(31 39)(32 34)```

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 6A 6B 8A 8B 8C 8D 8E 12A 12B 12C 16A 16B 16C 16D 24A 24B 24C 24D 24E 24F 48A ··· 48H order 1 2 2 2 2 3 4 4 6 6 8 8 8 8 8 12 12 12 16 16 16 16 24 24 24 24 24 24 48 ··· 48 size 1 1 2 24 24 2 2 2 2 4 2 2 4 24 24 2 2 4 4 4 4 4 2 2 2 2 4 4 4 ··· 4

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C4 S3 D4 D4 D6 D8 SD16 C4×S3 C3⋊D4 D12 D24 C24⋊C2 M5(2)⋊C2 M5(2)⋊S3 kernel M5(2)⋊S3 C24.C4 C3×M5(2) C2×D24 D24 M5(2) C24 C2×C12 C2×C8 C12 C2×C6 C8 C8 C2×C4 C4 C22 C3 C1 # reps 1 1 1 1 4 1 1 1 1 2 2 2 2 2 4 4 2 4

Matrix representation of M5(2)⋊S3 in GL4(𝔽97) generated by

 0 0 1 0 0 0 0 1 95 79 0 0 18 16 0 0
,
 1 0 0 0 0 1 0 0 0 0 96 0 0 0 0 96
,
 96 96 0 0 1 0 0 0 0 0 96 96 0 0 1 0
,
 1 0 0 0 96 96 0 0 0 0 2 18 0 0 16 95
`G:=sub<GL(4,GF(97))| [0,0,95,18,0,0,79,16,1,0,0,0,0,1,0,0],[1,0,0,0,0,1,0,0,0,0,96,0,0,0,0,96],[96,1,0,0,96,0,0,0,0,0,96,1,0,0,96,0],[1,96,0,0,0,96,0,0,0,0,2,16,0,0,18,95] >;`

M5(2)⋊S3 in GAP, Magma, Sage, TeX

`M_5(2)\rtimes S_3`
`% in TeX`

`G:=Group("M5(2):S3");`
`// GroupNames label`

`G:=SmallGroup(192,75);`
`// by ID`

`G=gap.SmallGroup(192,75);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,85,92,422,387,268,570,1684,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^16=b^2=c^3=d^2=1,b*a*b=a^9,a*c=c*a,d*a*d=a^11*b,b*c=c*b,b*d=d*b,d*c*d=c^-1>;`
`// generators/relations`

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