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## G = C23.C42order 128 = 27

### 2nd 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 — C23 — C23.C42
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C23.7Q8 — C23.C42
 Lower central C1 — C22 — C23 — C23.C42
 Upper central C1 — C22 — C23×C4 — C23.C42
 Jennings C1 — C2 — C22 — C23×C4 — C23.C42

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

Subgroups: 240 in 100 conjugacy classes, 34 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×5], C22 [×3], C22 [×12], C8 [×4], C2×C4 [×4], C2×C4 [×15], C23 [×3], C23 [×4], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×4], M4(2) [×4], C22×C4 [×6], C22×C4 [×4], C24, C2.C42, C22⋊C8 [×2], C22⋊C8 [×3], C2×C22⋊C4 [×2], C2×C4⋊C4, C2×M4(2) [×2], C2×M4(2), C23×C4, C23.7Q8, C24.4C4 [×2], C23.C42
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], C4.D4, C4.10D4, C23.9D4, C22.C42, M4(2)⋊4C4, C23.C42

Character table of C23.C42

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 1 1 2 2 4 4 2 2 2 2 4 4 8 8 8 8 8 8 8 8 8 8 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 -1 1 -i i i -i -1 1 i 1 i -i -i -1 linear of order 4 ρ6 1 1 1 1 -1 -1 -1 1 -1 -1 1 1 1 -1 i i -i -i i -i -1 i 1 -1 1 -i linear of order 4 ρ7 1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 -i i i -i 1 -1 -i -1 -i i i 1 linear of order 4 ρ8 1 1 1 1 -1 -1 -1 1 -1 -1 1 1 1 -1 i i -i -i -i i 1 -i -1 1 -1 i linear of order 4 ρ9 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 -1 1 -1 i i i -i -i -i i -i linear of order 4 ρ10 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 -1 1 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 -1 1 i -i -i i -1 1 -i 1 -i i i -1 linear of order 4 ρ12 1 1 1 1 -1 -1 -1 1 -1 -1 1 1 1 -1 -i -i i i i -i 1 i -1 1 -1 -i linear of order 4 ρ13 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 -1 1 -1 -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 -1 -1 1 -1 1 -i -i i i -i -i i i linear of order 4 ρ15 1 1 1 1 -1 -1 -1 1 -1 -1 1 1 1 -1 -i -i i i -i i -1 -i 1 -1 1 i linear of order 4 ρ16 1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 i -i -i i 1 -1 i -1 i -i -i 1 linear of order 4 ρ17 2 2 2 2 -2 -2 2 -2 2 2 2 2 -2 -2 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 2 -2 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 -2 2 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 2 2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ21 4 -4 4 -4 0 0 0 0 0 0 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ22 4 -4 4 -4 0 0 0 0 0 0 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ23 4 4 -4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C4.D4 ρ24 4 4 -4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C4.10D4, Schur index 2 ρ25 4 -4 -4 4 0 0 0 0 -4i 4i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from M4(2)⋊4C4 ρ26 4 -4 -4 4 0 0 0 0 4i -4i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from M4(2)⋊4C4

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

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

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

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

Matrix representation of C23.C42 in GL8(𝔽17)

 0 13 0 0 0 0 0 0 4 0 0 0 0 0 0 0 4 4 13 9 0 0 0 0 0 13 4 4 0 0 0 0 0 0 0 0 13 2 0 0 0 0 0 0 1 4 0 0 0 0 0 0 16 11 13 2 0 0 0 0 16 1 1 4
,
 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16
,
 0 0 1 0 0 0 0 0 1 1 16 15 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 16 0 0 0 0 0 0 0 0 12 0 15 0 0 0 0 0 1 1 0 15 0 0 0 0 12 0 5 0 0 0 0 0 15 0 16 16
,
 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 4 4 13 9 0 0 0 0 7 6 0 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 4 16 0 0 0 0 0 0 0 1 4 15 0 0 0 0 13 10 0 13

`G:=sub<GL(8,GF(17))| [0,4,4,0,0,0,0,0,13,0,4,13,0,0,0,0,0,0,13,4,0,0,0,0,0,0,9,4,0,0,0,0,0,0,0,0,13,1,16,16,0,0,0,0,2,4,11,1,0,0,0,0,0,0,13,1,0,0,0,0,0,0,2,4],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[0,1,0,1,0,0,0,0,0,1,1,0,0,0,0,0,1,16,0,0,0,0,0,0,0,15,0,16,0,0,0,0,0,0,0,0,12,1,12,15,0,0,0,0,0,1,0,0,0,0,0,0,15,0,5,16,0,0,0,0,0,15,0,16],[1,0,4,7,0,0,0,0,0,16,4,6,0,0,0,0,0,0,13,0,0,0,0,0,0,0,9,4,0,0,0,0,0,0,0,0,1,4,0,13,0,0,0,0,0,16,1,10,0,0,0,0,0,0,4,0,0,0,0,0,0,0,15,13] >;`

C23.C42 in GAP, Magma, Sage, TeX

`C_2^3.C_4^2`
`% in TeX`

`G:=Group("C2^3.C4^2");`
`// GroupNames label`

`G:=SmallGroup(128,37);`
`// by ID`

`G=gap.SmallGroup(128,37);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,723,570,136,2804]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^2=b^2=c^2=e^4=1,d^4=b,d*a*d^-1=a*b=b*a,e*a*e^-1=a*c=c*a,b*c=c*b,b*d=d*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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