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## G = C8.C24order 128 = 27

### 6th non-split extension by C8 of C24 acting via C24/C22=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C8.C24
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C2.C25 — C8.C24
 Lower central C1 — C2 — C4 — C8.C24
 Upper central C1 — C4 — C2×C4○D4 — C8.C24
 Jennings C1 — C2 — C2 — C4 — C8.C24

Generators and relations for C8.C24
G = < a,b,c,d,e | a8=b2=c2=1, d2=a6, e2=a4, bab=a-1, cac=a5, ad=da, ae=ea, bc=cb, dbd-1=a6b, be=eb, cd=dc, ce=ec, de=ed >

Subgroups: 1092 in 712 conjugacy classes, 426 normal (13 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4○D4, C22×C8, C2×M4(2), C8○D4, C2×D8, C2×SD16, C2×Q16, C4○D8, C8⋊C22, C8.C22, C2×C4○D4, C2×C4○D4, C2×C4○D4, 2+ 1+4, 2+ 1+4, 2- 1+4, 2- 1+4, C2×C8○D4, C2×C4○D8, D8⋊C22, D4○D8, D4○SD16, Q8○D8, C2.C25, C8.C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, C25, D4×C23, C8.C24

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 C8.C24 kernel C8.C24 C2×C8○D4 C2×C4○D8 D8⋊C22 D4○D8 D4○SD16 Q8○D8 C2.C25 C2×D4 C2×Q8 C4○D4 C1 # reps 1 1 6 6 4 8 4 2 3 1 4 4

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

 0 0 6 6 3 0 0 11 0 14 14 14 14 14 3 3
,
 16 0 15 0 1 0 1 16 0 0 1 0 16 16 16 0
,
 4 8 0 0 13 13 0 0 13 13 0 4 0 4 13 0
,
 11 11 0 0 3 0 0 0 0 3 14 3 14 14 14 14
,
 13 0 0 0 0 13 0 0 0 0 13 0 0 0 0 13
`G:=sub<GL(4,GF(17))| [0,3,0,14,0,0,14,14,6,0,14,3,6,11,14,3],[16,1,0,16,0,0,0,16,15,1,1,16,0,16,0,0],[4,13,13,0,8,13,13,4,0,0,0,13,0,0,4,0],[11,3,0,14,11,0,3,14,0,0,14,14,0,0,3,14],[13,0,0,0,0,13,0,0,0,0,13,0,0,0,0,13] >;`

C8.C24 in GAP, Magma, Sage, TeX

`C_8.C_2^4`
`% in TeX`

`G:=Group("C8.C2^4");`
`// GroupNames label`

`G:=SmallGroup(128,2316);`
`// by ID`

`G=gap.SmallGroup(128,2316);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,2,-2,-2,477,521,172,4037,2028,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^8=b^2=c^2=1,d^2=a^6,e^2=a^4,b*a*b=a^-1,c*a*c=a^5,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=a^6*b,b*e=e*b,c*d=d*c,c*e=e*c,d*e=e*d>;`
`// generators/relations`

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