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## G = C42.13C23order 128 = 27

### 13rd non-split extension by C42 of C23 acting faithfully

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.13C23
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — C2.C25 — C42.13C23
 Lower central C1 — C2 — C2×C4 — C42.13C23
 Upper central C1 — C2 — C22×C4 — C42.13C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.13C23

Generators and relations for C42.13C23
G = < a,b,c,d,e | a4=b4=c2=1, d2=e2=b2, cac=ab=ba, dad-1=a-1b2, eae-1=ab2, cbc=dbd-1=b-1, be=eb, dcd-1=b-1c, ce=ec, ede-1=b2d >

Subgroups: 668 in 352 conjugacy classes, 106 normal (18 characteristic)
C1, C2, C2 [×9], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×15], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×32], D4 [×4], D4 [×28], Q8 [×4], Q8 [×14], C23, C23 [×2], C23 [×6], C42 [×2], C22⋊C4 [×5], C4⋊C4 [×5], C2×C8 [×2], M4(2) [×4], M4(2) [×2], SD16 [×8], Q16 [×8], C22×C4, C22×C4 [×2], C22×C4 [×7], C2×D4, C2×D4 [×4], C2×D4 [×20], C2×Q8, C2×Q8 [×4], C2×Q8 [×11], C4○D4 [×8], C4○D4 [×36], C4.D4 [×2], C4.10D4 [×2], C4≀C2 [×8], C42⋊C2, C22⋊Q8 [×2], C22.D4 [×2], C4.4D4 [×2], C4⋊Q8 [×2], C2×M4(2) [×2], C2×SD16 [×2], C2×Q16 [×2], C8.C22 [×8], C8.C22 [×4], C22×Q8, C2×C4○D4, C2×C4○D4 [×2], C2×C4○D4 [×6], 2+ 1+4 [×2], 2+ 1+4 [×4], 2- 1+4 [×2], 2- 1+4 [×2], M4(2).8C22, C42⋊C22 [×2], D4.9D4 [×4], D4.10D4 [×4], C23.38C23, C2×C8.C22 [×2], C2.C25, C42.13C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C22×D4 [×3], C2×C22≀C2, C42.13C23

Character table of C42.13C23

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 8A 8B 8C 8D size 1 1 2 2 2 4 4 4 4 4 4 2 2 2 2 4 4 4 4 4 4 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 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 -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 -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 1 -1 -1 linear of order 2 ρ5 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 1 1 1 linear of order 2 ρ6 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 -1 1 1 linear of order 2 ρ7 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 -1 -1 -1 linear of order 2 ρ8 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 1 -1 -1 linear of order 2 ρ9 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 1 -1 1 linear of order 2 ρ10 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 -1 -1 1 linear of order 2 ρ11 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 -1 1 -1 linear of order 2 ρ12 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 1 1 -1 linear of order 2 ρ13 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 1 -1 1 linear of order 2 ρ14 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 -1 -1 1 linear of order 2 ρ15 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 -1 1 -1 linear of order 2 ρ16 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 1 1 -1 linear of order 2 ρ17 2 2 2 -2 -2 0 0 -2 0 -2 0 -2 2 -2 2 0 2 0 0 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 -2 -2 0 -2 0 0 0 2 2 -2 2 -2 2 0 0 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 -2 -2 2 0 0 -2 0 2 0 2 2 -2 -2 0 -2 0 0 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 -2 2 -2 2 0 0 -2 0 0 2 -2 -2 2 0 0 2 0 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 2 2 2 -2 -2 0 2 0 0 0 -2 2 -2 2 -2 -2 0 0 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ22 2 2 -2 2 -2 -2 0 0 2 0 0 2 -2 -2 2 0 0 -2 0 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ23 2 2 -2 -2 2 0 -2 0 0 0 -2 -2 -2 2 2 2 0 0 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ24 2 2 2 2 2 -2 0 0 2 0 0 -2 -2 -2 -2 0 0 2 0 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ25 2 2 2 2 2 2 0 0 -2 0 0 -2 -2 -2 -2 0 0 -2 0 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ26 2 2 -2 -2 2 0 0 2 0 -2 0 2 2 -2 -2 0 2 0 0 0 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ27 2 2 -2 -2 2 0 2 0 0 0 2 -2 -2 2 2 -2 0 0 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ28 2 2 2 -2 -2 0 0 2 0 2 0 -2 2 -2 2 0 -2 0 0 0 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ29 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

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

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

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

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

Matrix representation of C42.13C23 in GL8(𝔽17)

 3 13 13 4 10 7 0 10 3 13 4 13 10 7 0 7 4 13 3 13 0 7 10 7 13 4 3 13 0 10 10 7 13 14 14 3 1 16 0 1 0 0 1 16 0 0 0 8 3 14 13 14 0 16 1 16 16 1 0 0 0 9 0 0
,
 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0
,
 0 0 16 1 0 0 1 15 0 0 0 0 0 0 16 0 16 1 0 0 1 15 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 16 0 0 16 1 0 16 0 0 0 0 0 0 0 16 0 0 16 1 0 0
,
 4 13 3 13 0 7 10 7 13 4 3 13 0 10 10 7 3 13 13 4 10 7 0 10 3 13 4 13 10 7 0 7 3 14 13 14 0 16 1 16 16 1 0 0 0 9 0 0 13 14 14 3 1 16 0 1 0 0 1 16 0 0 0 8
,
 0 0 0 0 1 0 0 0 16 1 0 0 1 15 0 0 0 0 0 0 0 0 1 0 0 0 16 1 0 0 1 15 16 0 0 0 0 0 0 0 16 1 0 0 0 16 0 0 0 0 16 0 0 0 0 0 0 0 16 1 0 0 0 16

`G:=sub<GL(8,GF(17))| [3,3,4,13,13,0,3,16,13,13,13,4,14,0,14,1,13,4,3,3,14,1,13,0,4,13,13,13,3,16,14,0,10,10,0,0,1,0,0,0,7,7,7,10,16,0,16,9,0,0,10,10,0,0,1,0,10,7,7,7,1,8,16,0],[0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,0,16,0,0,0,0,0,0,0,1,0,0,0,16,16,16,0,0,0,0,0,0,0,1,0,0,0,16,16,0,0,0,0,1,16,0,0,0,16,0,0,15,0,0,0,0,1,1,16,0,0,0,16,0,0,15,0,0,0,0,1,0,0],[4,13,3,3,3,16,13,0,13,4,13,13,14,1,14,0,3,3,13,4,13,0,14,1,13,13,4,13,14,0,3,16,0,0,10,10,0,0,1,0,7,10,7,7,16,9,16,0,10,10,0,0,1,0,0,0,7,7,10,7,16,0,1,8],[0,16,0,0,16,16,0,0,0,1,0,0,0,1,0,0,0,0,0,16,0,0,16,16,0,0,0,1,0,0,0,1,1,1,0,0,0,0,0,0,0,15,0,0,0,16,0,0,0,0,1,1,0,0,0,0,0,0,0,15,0,0,0,16] >;`

C42.13C23 in GAP, Magma, Sage, TeX

`C_4^2._{13}C_2^3`
`% in TeX`

`G:=Group("C4^2.13C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1754);`
`// by ID`

`G=gap.SmallGroup(128,1754);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,2019,248,2804,1411,718,172,2028]);`
`// Polycyclic`

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

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