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G = C23.8M4(2)  order 128 = 27

4th non-split extension by C23 of M4(2) acting via M4(2)/C4=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C23.8M4(2)
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C2×C4×D4 — C23.8M4(2)
 Lower central C1 — C2 — C22 — C23.8M4(2)
 Upper central C1 — C2×C4 — C23×C4 — C23.8M4(2)
 Jennings C1 — C2 — C22 — C22×C4 — C23.8M4(2)

Generators and relations for C23.8M4(2)
G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, ab=ba, dad-1=eae=ac=ca, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede=bcd5 >

Subgroups: 348 in 166 conjugacy classes, 62 normal (26 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×7], C22 [×3], C22 [×4], C22 [×18], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×19], D4 [×8], C23, C23 [×8], C23 [×6], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×8], C22×C4 [×5], C22×C4 [×4], C22×C4 [×6], C2×D4 [×4], C2×D4 [×4], C24 [×2], C22⋊C8 [×4], C22⋊C8 [×2], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×4], C22×C8 [×2], C23×C4 [×2], C22×D4, C23⋊C8 [×2], C22.M4(2) [×2], C2×C22⋊C8 [×2], C2×C4×D4, C23.8M4(2)
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], C23⋊C4 [×2], C2×C22⋊C4, C22×C8, C2×M4(2), C2×C22⋊C8, C2×C23⋊C4, M4(2).8C22, C23.8M4(2)

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

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 2K 4A 4B 4C 4D 4E ··· 4J 4K ··· 4P 8A ··· 8P order 1 2 2 2 2 ··· 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 ··· 2 4 4 1 1 1 1 2 ··· 2 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C4 C8 D4 M4(2) C23⋊C4 M4(2).8C22 kernel C23.8M4(2) C23⋊C8 C22.M4(2) C2×C22⋊C8 C2×C4×D4 C2×C42 C2×C22⋊C4 C23×C4 C22×D4 C2×D4 C22×C4 C23 C4 C2 # reps 1 2 2 2 1 2 2 2 2 16 4 4 2 2

Matrix representation of C23.8M4(2) in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 11 16 0 0 0 11 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 10 11 16 0 0 0 6 7 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
,
 4 13 0 0 0 0 5 13 0 0 0 0 0 0 6 7 0 15 0 0 10 11 15 0 0 0 0 4 6 7 0 0 0 0 10 11
,
 1 0 0 0 0 0 2 16 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 16 0

`G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,11,0,0,0,1,11,0,0,0,0,0,16,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,10,6,0,0,0,1,11,7,0,0,0,0,16,0,0,0,0,0,0,16],[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],[4,5,0,0,0,0,13,13,0,0,0,0,0,0,6,10,0,0,0,0,7,11,4,0,0,0,0,15,6,10,0,0,15,0,7,11],[1,2,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0] >;`

C23.8M4(2) in GAP, Magma, Sage, TeX

`C_2^3._8M_4(2)`
`% in TeX`

`G:=Group("C2^3.8M4(2)");`
`// GroupNames label`

`G:=SmallGroup(128,191);`
`// by ID`

`G=gap.SmallGroup(128,191);`
`# by ID`

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

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

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