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## G = C42.10Q8order 128 = 27

### 10th non-split extension by C42 of Q8 acting via Q8/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.10Q8
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C42.12C4 — C42.10Q8
 Lower central C1 — C22 — C2×C4 — C42.10Q8
 Upper central C1 — C2×C4 — C2×C42 — C42.10Q8
 Jennings C1 — C22 — C22 — C2×C42 — C42.10Q8

Generators and relations for C42.10Q8
G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=a2b-1c2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b-1, dcd-1=ab2c3 >

Subgroups: 168 in 88 conjugacy classes, 42 normal (34 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×2], C4 [×7], C22 [×3], C22 [×2], C8 [×4], C2×C4 [×10], C2×C4 [×11], C23, C42 [×4], C42 [×2], C4⋊C4 [×4], C2×C8 [×4], M4(2) [×2], C22×C4 [×3], C22×C4 [×2], C2.C42 [×2], C4×C8, C22⋊C8, C4⋊C8 [×4], C4⋊C8, C2×C42, C2×C42, C2×C4⋊C4, C2×M4(2), C4×C4⋊C4, C4⋊M4(2), C42.12C4, C42.10Q8
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], D8, SD16 [×2], Q16, C2.C42, D4⋊C4 [×2], Q8⋊C4 [×2], C4≀C2 [×2], C4.Q8, C2.D8, C426C4, C22.4Q16, M4(2)⋊4C4, C42.10Q8

Smallest permutation representation of C42.10Q8
On 32 points
Generators in S32
```(1 31 15 23)(2 24 16 32)(3 25 9 17)(4 18 10 26)(5 27 11 19)(6 20 12 28)(7 29 13 21)(8 22 14 30)
(1 25 11 21)(2 26 12 22)(3 27 13 23)(4 28 14 24)(5 29 15 17)(6 30 16 18)(7 31 9 19)(8 32 10 20)
(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 31 28 15 12 23 20)(2 27 32 11 16 19 24 5)(3 4 25 26 9 10 17 18)(7 8 29 30 13 14 21 22)```

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E ··· 4J 4K ··· 4T 8A ··· 8H 8I 8J 8K 8L order 1 2 2 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 8 ··· 8 8 8 8 8 size 1 1 1 1 2 2 1 1 1 1 2 ··· 2 4 ··· 4 4 ··· 4 8 8 8 8

38 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + - + + - image C1 C2 C2 C2 C4 C4 D4 Q8 D4 D8 SD16 Q16 C4≀C2 M4(2)⋊4C4 kernel C42.10Q8 C4×C4⋊C4 C4⋊M4(2) C42.12C4 C2.C42 C4⋊C8 C42 C42 C22×C4 C2×C4 C2×C4 C2×C4 C4 C2 # reps 1 1 1 1 4 8 1 1 2 2 4 2 8 2

Matrix representation of C42.10Q8 in GL4(𝔽17) generated by

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

C42.10Q8 in GAP, Magma, Sage, TeX

`C_4^2._{10}Q_8`
`% in TeX`

`G:=Group("C4^2.10Q8");`
`// GroupNames label`

`G:=SmallGroup(128,35);`
`// by ID`

`G=gap.SmallGroup(128,35);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,723,184,248,3924,242]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2*b^2,d^2=a^2*b^-1*c^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=a*b^2*c^3>;`
`// generators/relations`

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