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## G = C23.15C42order 128 = 27

### 10th non-split extension by C23 of C42 acting via C42/C22=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C23.15C42
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C2×C42⋊C2 — C23.15C42
 Lower central C1 — C2 — C22 — C23.15C42
 Upper central C1 — C2×C4 — C23×C4 — C23.15C42
 Jennings C1 — C2 — C2 — C22×C4 — C23.15C42

Generators and relations for C23.15C42
G = < a,b,c,d,e | a2=b2=c2=e4=1, d4=c, ab=ba, eae-1=ac=ca, ad=da, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=bcd >

Subgroups: 308 in 194 conjugacy classes, 108 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×8], C4 [×4], C22 [×3], C22 [×4], C22 [×10], C8 [×8], C2×C4 [×4], C2×C4 [×24], C2×C4 [×8], C23, C23 [×6], C23 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×12], M4(2) [×8], M4(2) [×12], C22×C4 [×2], C22×C4 [×16], C24, C2×C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×4], C22×C8 [×2], C2×M4(2) [×12], C2×M4(2) [×6], C23×C4, C22.C42 [×4], C2×C42⋊C2, C22×M4(2) [×2], C23.15C42
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×6], Q8 [×2], C23, C42 [×4], C22⋊C4 [×12], C4⋊C4 [×12], C22×C4 [×3], C2×D4 [×3], C2×Q8, C2.C42 [×8], C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4 [×3], C2×C2.C42, M4(2).8C22 [×2], C23.15C42

Smallest permutation representation of C23.15C42
On 32 points
Generators in S32
```(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)
(1 31)(2 28)(3 25)(4 30)(5 27)(6 32)(7 29)(8 26)(9 21)(10 18)(11 23)(12 20)(13 17)(14 22)(15 19)(16 24)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 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 17 31 13)(2 10 32 22)(3 23 25 11)(4 16 26 20)(5 21 27 9)(6 14 28 18)(7 19 29 15)(8 12 30 24)```

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 4 type + + + + + - image C1 C2 C2 C2 C4 C4 C4 D4 Q8 M4(2).8C22 kernel C23.15C42 C22.C42 C2×C42⋊C2 C22×M4(2) C2×C42 C2×C22⋊C4 C2×M4(2) C22×C4 C22×C4 C2 # reps 1 4 1 2 4 4 16 6 2 4

Matrix representation of C23.15C42 in GL6(𝔽17)

 1 0 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 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 0 0 0 0 0 1 0 0 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
,
 0 16 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 13 0 0 0 0 0 0 4 0 0
,
 4 0 0 0 0 0 0 13 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0

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

C23.15C42 in GAP, Magma, Sage, TeX

`C_2^3._{15}C_4^2`
`% in TeX`

`G:=Group("C2^3.15C4^2");`
`// GroupNames label`

`G:=SmallGroup(128,474);`
`// by ID`

`G=gap.SmallGroup(128,474);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,352,2019,1411,172]);`
`// Polycyclic`

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

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