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## G = C42.62D4order 128 = 27

### 44th 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 — C42.62D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C23.37C23 — C42.62D4
 Lower central C1 — C22 — C2×C4 — C42.62D4
 Upper central C1 — C22 — C2×C42 — C42.62D4
 Jennings C1 — C2 — C22 — C22×C4 — C42.62D4

Generators and relations for C42.62D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2b-1, ab=ba, cac-1=ab2, dad-1=a-1b2, cbc-1=a2b, bd=db, dcd-1=b-1c-1 >

Subgroups: 220 in 110 conjugacy classes, 52 normal (28 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×10], C22 [×3], C22 [×2], C8 [×2], C2×C4 [×6], C2×C4 [×4], C2×C4 [×12], Q8 [×4], C23, C42 [×4], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×9], C2×C8 [×2], C22×C4 [×3], C22×C4 [×2], C2×Q8 [×2], C2×Q8, C2.C42 [×2], C2.C42, C4×C8, C22⋊C8 [×2], C4⋊C8, C2×C42, C2×C4⋊C4, C42⋊C2, C4×Q8 [×2], C4×Q8, C22⋊Q8 [×2], C22⋊Q8, C42.C2, C4⋊Q8 [×2], C23.31D4 [×4], C428C4, C42.12C4, C23.37C23, C42.62D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], SD16 [×2], Q16 [×2], C22×C4, C2×D4 [×2], Q8⋊C4 [×4], C2×C22⋊C4, C2×SD16, C2×Q16, C23.C23, C2×Q8⋊C4, C42⋊C22, C42.62D4

Smallest permutation representation of C42.62D4
On 32 points
Generators in S32
```(1 15 32 17)(2 22 25 12)(3 9 26 19)(4 24 27 14)(5 11 28 21)(6 18 29 16)(7 13 30 23)(8 20 31 10)
(1 30 5 26)(2 31 6 27)(3 32 7 28)(4 25 8 29)(9 17 13 21)(10 18 14 22)(11 19 15 23)(12 20 16 24)
(1 17 28 11)(2 14)(3 9 30 23)(4 18)(5 21 32 15)(6 10)(7 13 26 19)(8 22)(12 31)(16 27)(20 29)(24 25)
(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)| (1,15,32,17)(2,22,25,12)(3,9,26,19)(4,24,27,14)(5,11,28,21)(6,18,29,16)(7,13,30,23)(8,20,31,10), (1,30,5,26)(2,31,6,27)(3,32,7,28)(4,25,8,29)(9,17,13,21)(10,18,14,22)(11,19,15,23)(12,20,16,24), (1,17,28,11)(2,14)(3,9,30,23)(4,18)(5,21,32,15)(6,10)(7,13,26,19)(8,22)(12,31)(16,27)(20,29)(24,25), (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( (1,15,32,17)(2,22,25,12)(3,9,26,19)(4,24,27,14)(5,11,28,21)(6,18,29,16)(7,13,30,23)(8,20,31,10), (1,30,5,26)(2,31,6,27)(3,32,7,28)(4,25,8,29)(9,17,13,21)(10,18,14,22)(11,19,15,23)(12,20,16,24), (1,17,28,11)(2,14)(3,9,30,23)(4,18)(5,21,32,15)(6,10)(7,13,26,19)(8,22)(12,31)(16,27)(20,29)(24,25), (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([(1,15,32,17),(2,22,25,12),(3,9,26,19),(4,24,27,14),(5,11,28,21),(6,18,29,16),(7,13,30,23),(8,20,31,10)], [(1,30,5,26),(2,31,6,27),(3,32,7,28),(4,25,8,29),(9,17,13,21),(10,18,14,22),(11,19,15,23),(12,20,16,24)], [(1,17,28,11),(2,14),(3,9,30,23),(4,18),(5,21,32,15),(6,10),(7,13,26,19),(8,22),(12,31),(16,27),(20,29),(24,25)], [(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 4J 4K ··· 4R 8A ··· 8H order 1 2 2 2 2 2 4 ··· 4 4 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 2 2 ··· 2 4 4 8 ··· 8 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + - image C1 C2 C2 C2 C2 C4 C4 D4 D4 SD16 Q16 C23.C23 C42⋊C22 kernel C42.62D4 C23.31D4 C42⋊8C4 C42.12C4 C23.37C23 C4×Q8 C4⋊Q8 C42 C22×C4 C2×C4 C2×C4 C2 C2 # reps 1 4 1 1 1 4 4 2 2 4 4 2 2

Matrix representation of C42.62D4 in GL6(𝔽17)

 13 15 0 0 0 0 0 4 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 16 16 16 15 0 0 0 0 0 1
,
 4 2 0 0 0 0 0 13 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 1 9 0 0 0 0 13 16 0 0 0 0 0 0 0 16 0 0 0 0 1 0 0 0 0 0 4 4 4 8 0 0 6 7 13 13
,
 2 11 0 0 0 0 0 9 0 0 0 0 0 0 4 4 4 8 0 0 0 0 4 0 0 0 0 16 0 0 0 0 6 7 13 13

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

C42.62D4 in GAP, Magma, Sage, TeX

`C_4^2._{62}D_4`
`% in TeX`

`G:=Group("C4^2.62D4");`
`// GroupNames label`

`G:=SmallGroup(128,250);`
`// by ID`

`G=gap.SmallGroup(128,250);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1430,387,184,1123,1018,248,1971]);`
`// Polycyclic`

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

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