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## G = C42.4Q8order 128 = 27

### 4th 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 — C42.4Q8
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C4⋊C4 — C42.4Q8
 Lower central C1 — C2 — C2×C4 — C42.4Q8
 Upper central C1 — C22 — C2×C42 — C42.4Q8
 Jennings C1 — C22 — C22 — C2×C42 — C42.4Q8

Generators and relations for C42.4Q8
G = < a,b,c,d | a4=b4=c4=1, d2=bc2, ab=ba, ac=ca, dad-1=a-1b2, cbc-1=a2b-1, dbd-1=a2b, dcd-1=a-1b2c-1 >

Subgroups: 160 in 82 conjugacy classes, 34 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×9], C22 [×3], C22 [×2], C8 [×4], C2×C4 [×6], C2×C4 [×13], C23, C42 [×4], C42 [×2], C4⋊C4 [×4], C2×C8 [×4], C22×C4 [×3], C22×C4 [×2], C2.C42, C8⋊C4 [×2], C8⋊C4, C22⋊C8 [×2], C4⋊C8 [×2], C4⋊C8, C2×C42, C2×C42 [×2], C2×C4⋊C4, C4×C4⋊C4, C42.6C4 [×2], C42.4Q8
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, C4.D4, C4.10D4, C4≀C2 [×2], C4.9C42, C426C4, C22.C42, C42.4Q8

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4H 4I ··· 4R 8A ··· 8H order 1 2 2 2 2 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 2 2 ··· 2 4 ··· 4 8 ··· 8

32 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 4 type + + + + - + + - image C1 C2 C2 C4 C4 C4 D4 Q8 D4 C4≀C2 C4.D4 C4.10D4 C4.9C42 kernel C42.4Q8 C4×C4⋊C4 C42.6C4 C8⋊C4 C4⋊C8 C2×C42 C42 C42 C22×C4 C4 C22 C22 C2 # reps 1 1 2 4 4 4 1 1 2 8 1 1 2

Matrix representation of C42.4Q8 in GL6(𝔽17)

 4 0 0 0 0 0 0 4 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 4 0 0 0 0 0 4 13 0 0 0 0 0 0 16 0 0 0 0 0 2 1 0 0 0 0 0 0 1 0 0 0 0 0 15 16
,
 4 0 0 0 0 0 10 1 0 0 0 0 0 0 1 1 0 0 0 0 15 16 0 0 0 0 0 0 4 4 0 0 0 0 9 13
,
 1 15 0 0 0 0 11 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 15 16 0 0

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

C42.4Q8 in GAP, Magma, Sage, TeX

`C_4^2._4Q_8`
`% in TeX`

`G:=Group("C4^2.4Q8");`
`// GroupNames label`

`G:=SmallGroup(128,17);`
`// by ID`

`G=gap.SmallGroup(128,17);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,520,1018,136,3924]);`
`// Polycyclic`

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

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