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## G = M5(2)⋊12C22order 128 = 27

### 8th semidirect product of M5(2) and C22 acting via C22/C2=C2

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — M5(2)⋊12C22
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C22×C8 — C2×C8○D4 — M5(2)⋊12C22
 Lower central C1 — C2 — C4 — M5(2)⋊12C22
 Upper central C1 — C8 — C22×C8 — M5(2)⋊12C22
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — M5(2)⋊12C22

Generators and relations for M5(2)⋊12C22
G = < a,b,c,d | a16=b2=c2=d2=1, bab=dad=a9, cac=a5b, cbc=a8b, bd=db, cd=dc >

Subgroups: 172 in 110 conjugacy classes, 58 normal (36 characteristic)
C1, C2, C2 [×5], C4 [×4], C4 [×2], C22 [×3], C22 [×5], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×5], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C16 [×4], C2×C8 [×6], C2×C8 [×5], M4(2) [×2], M4(2) [×5], C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], C2×C16 [×2], C2×C16, M5(2) [×2], M5(2) [×3], C22×C8, C22×C8, C2×M4(2), C2×M4(2), C8○D4 [×4], C8○D4 [×2], C2×C4○D4, D4.C8 [×4], C2×M5(2) [×2], C2×C8○D4, M5(2)⋊12C22
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4 [×2], C22⋊C8 [×4], C2×C22⋊C4, C22×C8, C2×M4(2), C2×C22⋊C8, M5(2)⋊12C22

Smallest permutation representation of M5(2)⋊12C22
On 32 points
Generators in S32
```(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)
(1 25)(2 18)(3 27)(4 20)(5 29)(6 22)(7 31)(8 24)(9 17)(10 26)(11 19)(12 28)(13 21)(14 30)(15 23)(16 32)
(2 22)(4 24)(6 26)(8 28)(10 30)(12 32)(14 18)(16 20)(17 25)(19 27)(21 29)(23 31)
(2 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)```

`G:=sub<Sym(32)| (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), (1,25)(2,18)(3,27)(4,20)(5,29)(6,22)(7,31)(8,24)(9,17)(10,26)(11,19)(12,28)(13,21)(14,30)(15,23)(16,32), (2,22)(4,24)(6,26)(8,28)(10,30)(12,32)(14,18)(16,20)(17,25)(19,27)(21,29)(23,31), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32)>;`

`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), (1,25)(2,18)(3,27)(4,20)(5,29)(6,22)(7,31)(8,24)(9,17)(10,26)(11,19)(12,28)(13,21)(14,30)(15,23)(16,32), (2,22)(4,24)(6,26)(8,28)(10,30)(12,32)(14,18)(16,20)(17,25)(19,27)(21,29)(23,31), (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32) );`

`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)], [(1,25),(2,18),(3,27),(4,20),(5,29),(6,22),(7,31),(8,24),(9,17),(10,26),(11,19),(12,28),(13,21),(14,30),(15,23),(16,32)], [(2,22),(4,24),(6,26),(8,28),(10,30),(12,32),(14,18),(16,20),(17,25),(19,27),(21,29),(23,31)], [(2,10),(4,12),(6,14),(8,16),(18,26),(20,28),(22,30),(24,32)])`

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 8A 8B 8C 8D 8E ··· 8J 8K 8L 8M 8N 16A ··· 16P order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 8 8 8 8 8 ··· 8 8 8 8 8 16 ··· 16 size 1 1 2 2 2 4 4 1 1 2 2 2 4 4 1 1 1 1 2 ··· 2 4 4 4 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 4 type + + + + + image C1 C2 C2 C2 C4 C4 C4 C8 C8 C8 D4 M4(2) M4(2) M5(2)⋊12C22 kernel M5(2)⋊12C22 D4.C8 C2×M5(2) C2×C8○D4 C2×M4(2) C8○D4 C2×C4○D4 C2×D4 C2×Q8 C4○D4 C2×C8 C2×C4 C23 C1 # reps 1 4 2 1 2 4 2 4 4 8 4 2 2 4

Matrix representation of M5(2)⋊12C22 in GL4(𝔽17) generated by

 0 0 2 13 0 0 14 15 13 8 0 0 6 4 0 0
,
 2 13 0 0 5 15 0 0 0 0 2 13 0 0 5 15
,
 16 0 0 0 16 1 0 0 0 0 9 16 0 0 12 8
,
 1 0 0 0 0 1 0 0 0 0 16 0 0 0 0 16
`G:=sub<GL(4,GF(17))| [0,0,13,6,0,0,8,4,2,14,0,0,13,15,0,0],[2,5,0,0,13,15,0,0,0,0,2,5,0,0,13,15],[16,16,0,0,0,1,0,0,0,0,9,12,0,0,16,8],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16] >;`

M5(2)⋊12C22 in GAP, Magma, Sage, TeX

`M_5(2)\rtimes_{12}C_2^2`
`% in TeX`

`G:=Group("M5(2):12C2^2");`
`// GroupNames label`

`G:=SmallGroup(128,849);`
`// by ID`

`G=gap.SmallGroup(128,849);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,1430,1018,248,1411,102,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^16=b^2=c^2=d^2=1,b*a*b=d*a*d=a^9,c*a*c=a^5*b,c*b*c=a^8*b,b*d=d*b,c*d=d*c>;`
`// generators/relations`

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