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## G = C42.102D4order 128 = 27

### 84th non-split extension by C42 of D4 acting via D4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.102D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C4×C4○D4 — C42.102D4
 Lower central C1 — C2 — C2×C4 — C42.102D4
 Upper central C1 — C2×C4 — C2×C42 — C42.102D4
 Jennings C1 — C2 — C2 — C22×C4 — C42.102D4

Generators and relations for C42.102D4
G = < a,b,c,d | a4=b4=c4=1, d2=b, ab=ba, cac-1=a-1b2, dad-1=a-1, bc=cb, bd=db, dcd-1=bc-1 >

Subgroups: 292 in 156 conjugacy classes, 62 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×2], C4 [×11], C22 [×3], C22 [×6], C8 [×2], C2×C4 [×10], C2×C4 [×22], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C42 [×4], C42 [×7], C22⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×6], C2×C8 [×2], M4(2) [×4], C22×C4 [×3], C22×C4 [×5], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], C2.C42, C4≀C2 [×4], C4⋊C8 [×2], C2×C42, C2×C42 [×2], C2×C42, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4×Q8 [×2], C2×M4(2) [×2], C2×C4○D4, C426C4 [×2], C4×C4⋊C4, C2×C4≀C2 [×2], C4⋊M4(2), C4×C4○D4, C42.102D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C22⋊C4 [×4], C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C4≀C2 [×2], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], C23.7Q8, C2×C4≀C2, C42⋊C22, C42.102D4

Smallest permutation representation of C42.102D4
On 32 points
Generators in S32
```(1 24 26 9)(2 10 27 17)(3 18 28 11)(4 12 29 19)(5 20 30 13)(6 14 31 21)(7 22 32 15)(8 16 25 23)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)
(1 11)(2 14 6 10)(3 13)(4 16 8 12)(5 15)(7 9)(17 27 21 31)(18 26)(19 29 23 25)(20 28)(22 30)(24 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)```

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4J 4K ··· 4Z 8A 8B 8C 8D order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 8 8 8 8 size 1 1 1 1 2 2 4 4 1 1 1 1 2 ··· 2 4 ··· 4 8 8 8 8

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 C4 D4 D4 D4 Q8 C4○D4 C4≀C2 C42⋊C22 kernel C42.102D4 C42⋊6C4 C4×C4⋊C4 C2×C4≀C2 C4⋊M4(2) C4×C4○D4 C4×D4 C4×Q8 C42 C22×C4 C4○D4 C4○D4 C2×C4 C4 C2 # reps 1 2 1 2 1 1 4 4 2 2 2 2 4 8 2

Matrix representation of C42.102D4 in GL4(𝔽17) generated by

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

C42.102D4 in GAP, Magma, Sage, TeX

`C_4^2._{102}D_4`
`% in TeX`

`G:=Group("C4^2.102D4");`
`// GroupNames label`

`G:=SmallGroup(128,538);`
`// by ID`

`G=gap.SmallGroup(128,538);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,2804,718,172,124]);`
`// Polycyclic`

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

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