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## G = C24.5C8order 128 = 27

### 2nd non-split extension by C24 of C8 acting via C8/C4=C2

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C24.5C8
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C22×C8 — C23×C8 — C24.5C8
 Lower central C1 — C22 — C24.5C8
 Upper central C1 — C2×C8 — C24.5C8
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C24.5C8

Generators and relations for C24.5C8
G = < a,b,c,d,e | a2=b2=c2=d2=1, e8=d, ab=ba, eae-1=ac=ca, ad=da, bc=cb, ebe-1=bd=db, cd=dc, ce=ec, de=ed >

Subgroups: 196 in 130 conjugacy classes, 64 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C16, C2×C8, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C24, C2×C16, M5(2), C22×C8, C22×C8, C22×C8, C23×C4, C22⋊C16, C2×M5(2), C23×C8, C24.5C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C22⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C22⋊C8, M5(2), C2×C22⋊C4, C22×C8, C2×M4(2), C2×C22⋊C8, C2×M5(2), C24.5C8

Smallest permutation representation of C24.5C8
On 32 points
Generators in S32
```(2 24)(4 26)(6 28)(8 30)(10 32)(12 18)(14 20)(16 22)
(1 31)(2 24)(3 17)(4 26)(5 19)(6 28)(7 21)(8 30)(9 23)(10 32)(11 25)(12 18)(13 27)(14 20)(15 29)(16 22)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 17)(12 18)(13 19)(14 20)(15 21)(16 22)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 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)```

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A 4B 4C 4D 4E ··· 4J 8A ··· 8H 8I ··· 8T 16A ··· 16P order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 8 ··· 8 8 ··· 8 16 ··· 16 size 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 type + + + + + image C1 C2 C2 C2 C4 C4 C8 C8 D4 M4(2) M5(2) kernel C24.5C8 C22⋊C16 C2×M5(2) C23×C8 C22×C8 C23×C4 C22×C4 C24 C2×C8 C2×C4 C22 # reps 1 4 2 1 6 2 12 4 4 4 16

Matrix representation of C24.5C8 in GL4(𝔽17) generated by

 1 0 0 0 0 1 0 0 0 0 1 0 0 0 8 16
,
 16 0 0 0 0 1 0 0 0 0 16 0 0 0 0 16
,
 1 0 0 0 0 1 0 0 0 0 16 0 0 0 0 16
,
 16 0 0 0 0 16 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 9 0 0 0 0 0 9 2 0 0 11 8
`G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,1,8,0,0,0,16],[16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[0,9,0,0,1,0,0,0,0,0,9,11,0,0,2,8] >;`

C24.5C8 in GAP, Magma, Sage, TeX

`C_2^4._5C_8`
`% in TeX`

`G:=Group("C2^4.5C8");`
`// GroupNames label`

`G:=SmallGroup(128,844);`
`// by ID`

`G=gap.SmallGroup(128,844);`
`# by ID`

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

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

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