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## G = C23.2C42order 128 = 27

### 2nd non-split extension by C23 of C42 acting via C42/C4=C4

p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C23.2C42
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — (C22×C8)⋊C2 — C23.2C42
 Lower central C1 — C2 — C22 — C23 — C23.2C42
 Upper central C1 — C4 — C2×C4 — C2×C4○D4 — C23.2C42
 Jennings C1 — C2 — C22 — C2×C4○D4 — C23.2C42

Generators and relations for C23.2C42
G = < a,b,c,d,e | a2=b2=c2=d4=1, e4=c, ab=ba, ac=ca, dad-1=abc, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=abcd >

Subgroups: 208 in 83 conjugacy classes, 30 normal (16 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×7], C22, C22 [×6], C8 [×2], C2×C4 [×2], C2×C4 [×2], C2×C4 [×10], D4 [×3], Q8, C23, C23 [×2], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×3], M4(2), C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×2], C22⋊C8 [×2], C23⋊C4 [×4], C23⋊C4 [×2], C42⋊C2 [×2], C22×C8, C2×M4(2), C2×C4○D4, (C22×C8)⋊C2, C23.C23 [×2], C23.2C42
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, C23⋊C4 [×2], C23.9D4, C23.2C42

Character table of C23.2C42

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 8A 8B 8C 8D 8E 8F size 1 1 2 4 4 4 1 1 2 4 4 4 8 8 8 8 8 8 8 8 4 4 4 4 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 1 1 1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 linear of order 2 ρ5 1 1 1 -1 1 -1 1 1 1 -1 -1 1 i -i i i -i -i -i i 1 1 1 1 -1 -1 linear of order 4 ρ6 1 1 1 1 1 -1 -1 -1 -1 -1 1 -1 -i 1 1 -1 -1 i -i i i -i i -i i -i linear of order 4 ρ7 1 1 1 -1 1 1 -1 -1 -1 1 -1 -1 -1 -i i -i i -1 1 1 -i i -i i i -i linear of order 4 ρ8 1 1 1 1 1 -1 -1 -1 -1 -1 1 -1 i 1 1 -1 -1 -i i -i -i i -i i -i i linear of order 4 ρ9 1 1 1 -1 1 -1 1 1 1 -1 -1 1 -i -i i i -i i i -i -1 -1 -1 -1 1 1 linear of order 4 ρ10 1 1 1 1 1 -1 -1 -1 -1 -1 1 -1 -i -1 -1 1 1 i -i i -i i -i i -i i linear of order 4 ρ11 1 1 1 -1 1 -1 1 1 1 -1 -1 1 i i -i -i i -i -i i -1 -1 -1 -1 1 1 linear of order 4 ρ12 1 1 1 -1 1 -1 1 1 1 -1 -1 1 -i i -i -i i i i -i 1 1 1 1 -1 -1 linear of order 4 ρ13 1 1 1 1 1 -1 -1 -1 -1 -1 1 -1 i -1 -1 1 1 -i i -i i -i i -i i -i linear of order 4 ρ14 1 1 1 -1 1 1 -1 -1 -1 1 -1 -1 -1 i -i i -i -1 1 1 i -i i -i -i i linear of order 4 ρ15 1 1 1 -1 1 1 -1 -1 -1 1 -1 -1 1 -i i -i i 1 -1 -1 i -i i -i -i i linear of order 4 ρ16 1 1 1 -1 1 1 -1 -1 -1 1 -1 -1 1 i -i i -i 1 -1 -1 -i i -i i i -i linear of order 4 ρ17 2 2 2 -2 -2 2 2 2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 2 -2 -2 2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 2 -2 2 -2 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 -2 -2 -2 -2 -2 -2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ21 4 4 -4 0 0 0 4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ22 4 4 -4 0 0 0 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ23 4 -4 0 0 0 0 4i -4i 0 0 0 0 0 0 0 0 0 0 0 0 2ζ85 2ζ87 2ζ8 2ζ83 0 0 complex faithful ρ24 4 -4 0 0 0 0 4i -4i 0 0 0 0 0 0 0 0 0 0 0 0 2ζ8 2ζ83 2ζ85 2ζ87 0 0 complex faithful ρ25 4 -4 0 0 0 0 -4i 4i 0 0 0 0 0 0 0 0 0 0 0 0 2ζ87 2ζ85 2ζ83 2ζ8 0 0 complex faithful ρ26 4 -4 0 0 0 0 -4i 4i 0 0 0 0 0 0 0 0 0 0 0 0 2ζ83 2ζ8 2ζ87 2ζ85 0 0 complex faithful

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

Matrix representation of C23.2C42 in GL4(𝔽17) generated by

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

C23.2C42 in GAP, Magma, Sage, TeX

`C_2^3._2C_4^2`
`% in TeX`

`G:=Group("C2^3.2C4^2");`
`// GroupNames label`

`G:=SmallGroup(128,123);`
`// by ID`

`G=gap.SmallGroup(128,123);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,758,570,521,172,4037]);`
`// Polycyclic`

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

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