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## G = C23⋊M4(2)  order 128 = 27

### The semidirect product of C23 and M4(2) acting via M4(2)/C4=C4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C23⋊M4(2)
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C2×C4×D4 — C23⋊M4(2)
 Lower central C1 — C2 — C23 — C23⋊M4(2)
 Upper central C1 — C22 — C23×C4 — C23⋊M4(2)
 Jennings C1 — C2 — C22 — C22×C4 — C23⋊M4(2)

Generators and relations for C23⋊M4(2)
G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, dad-1=ab=ba, eae=ac=ca, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede=d5 >

Subgroups: 348 in 158 conjugacy classes, 50 normal (26 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×7], C22 [×3], C22 [×20], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×19], D4 [×8], C23, C23 [×4], C23 [×8], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], M4(2) [×4], C22×C4 [×5], C22×C4 [×4], C22×C4 [×6], C2×D4 [×6], C24 [×2], C22⋊C8 [×4], C22⋊C8 [×2], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×4], C2×M4(2) [×2], C23×C4 [×2], C22×D4, C23⋊C8 [×2], C22.M4(2) [×2], C24.4C4 [×2], C2×C4×D4, C23⋊M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4 [×2], C23⋊C4 [×2], C2×C22⋊C4, C2×M4(2) [×2], C24.4C4, C2×C23⋊C4, M4(2).8C22, C23⋊M4(2)

Smallest permutation representation of C23⋊M4(2)
On 32 points
Generators in S32
```(1 5)(2 6)(3 29)(4 30)(7 25)(8 26)(9 19)(10 20)(11 15)(12 16)(13 23)(14 24)(17 21)(18 22)(27 31)(28 32)
(2 32)(4 26)(6 28)(8 30)(10 24)(12 18)(14 20)(16 22)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 23)(10 24)(11 17)(12 18)(13 19)(14 20)(15 21)(16 22)
(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 13)(2 10)(3 15)(4 12)(5 9)(6 14)(7 11)(8 16)(17 29)(18 26)(19 31)(20 28)(21 25)(22 30)(23 27)(24 32)```

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A ··· 4H 4I ··· 4N 8A ··· 8H order 1 2 2 2 2 2 2 2 2 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 2 4 4 4 4 2 ··· 2 4 ··· 4 8 ··· 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 4 4 type + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C4 D4 M4(2) M4(2) C23⋊C4 M4(2).8C22 kernel C23⋊M4(2) C23⋊C8 C22.M4(2) C24.4C4 C2×C4×D4 C2×C42 C2×C22⋊C4 C23×C4 C22×D4 C22×C4 C2×C4 C23 C4 C2 # reps 1 2 2 2 1 2 2 2 2 4 4 4 2 2

Matrix representation of C23⋊M4(2) in GL6(𝔽17)

 16 2 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 11 1 0 0 0 8 0 0 16
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 12 6 16 0 0 0 8 10 0 16
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 1 12 0 0 0 0 1 16 0 0 0 0 0 0 9 7 0 2 0 0 5 11 2 0 0 0 10 7 6 12 0 0 10 7 10 8
,
 16 2 0 0 0 0 0 1 0 0 0 0 0 0 0 16 0 0 0 0 16 0 0 0 0 0 1 16 0 16 0 0 1 16 16 0

`G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,2,1,0,0,0,0,0,0,1,0,0,8,0,0,0,16,11,0,0,0,0,0,1,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,12,8,0,0,0,1,6,10,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,1,0,0,0,0,12,16,0,0,0,0,0,0,9,5,10,10,0,0,7,11,7,7,0,0,0,2,6,10,0,0,2,0,12,8],[16,0,0,0,0,0,2,1,0,0,0,0,0,0,0,16,1,1,0,0,16,0,16,16,0,0,0,0,0,16,0,0,0,0,16,0] >;`

C23⋊M4(2) in GAP, Magma, Sage, TeX

`C_2^3\rtimes M_4(2)`
`% in TeX`

`G:=Group("C2^3:M4(2)");`
`// GroupNames label`

`G:=SmallGroup(128,197);`
`// by ID`

`G=gap.SmallGroup(128,197);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,352,1123,851,172]);`
`// Polycyclic`

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

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