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## G = C42.61D4order 128 = 27

### 43rd 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 — C42.61D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C22.26C24 — C42.61D4
 Lower central C1 — C22 — C2×C4 — C42.61D4
 Upper central C1 — C22 — C2×C42 — C42.61D4
 Jennings C1 — C2 — C22 — C22×C4 — C42.61D4

Generators and relations for C42.61D4
G = < a,b,c,d | a4=b4=c4=1, d2=b-1, ab=ba, cac-1=ab2, dad-1=a-1b2, cbc-1=a2b, bd=db, dcd-1=b-1c-1 >

Subgroups: 316 in 132 conjugacy classes, 52 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×8], C22 [×3], C22 [×8], C8 [×2], C2×C4 [×6], C2×C4 [×4], C2×C4 [×14], D4 [×10], Q8 [×2], C23, C23 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], C22×C4 [×3], C22×C4 [×4], C2×D4 [×2], C2×D4 [×4], C2×Q8, C4○D4 [×4], C2.C42 [×2], C2.C42, C4×C8, C22⋊C8 [×2], C4⋊C8, C2×C42, C2×C4⋊C4, C4×D4 [×2], C4×D4, C4⋊D4 [×2], C4⋊D4, C4.4D4, C41D4, C4⋊Q8, C2×C4○D4, C22.SD16 [×4], C428C4, C42.12C4, C22.26C24, C42.61D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], D4⋊C4 [×4], C2×C22⋊C4, C2×D8, C2×SD16, C23.C23, C2×D4⋊C4, C42⋊C22, C42.61D4

Smallest permutation representation of C42.61D4
On 32 points
Generators in S32
```(1 9 30 17)(2 22 31 14)(3 11 32 19)(4 24 25 16)(5 13 26 21)(6 18 27 10)(7 15 28 23)(8 20 29 12)
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)
(1 17 26 13)(2 24)(3 11 28 23)(4 10)(5 21 30 9)(6 20)(7 15 32 19)(8 14)(12 27)(16 31)(18 25)(22 29)
(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)| (1,9,30,17)(2,22,31,14)(3,11,32,19)(4,24,25,16)(5,13,26,21)(6,18,27,10)(7,15,28,23)(8,20,29,12), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28), (1,17,26,13)(2,24)(3,11,28,23)(4,10)(5,21,30,9)(6,20)(7,15,32,19)(8,14)(12,27)(16,31)(18,25)(22,29), (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( (1,9,30,17)(2,22,31,14)(3,11,32,19)(4,24,25,16)(5,13,26,21)(6,18,27,10)(7,15,28,23)(8,20,29,12), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28), (1,17,26,13)(2,24)(3,11,28,23)(4,10)(5,21,30,9)(6,20)(7,15,32,19)(8,14)(12,27)(16,31)(18,25)(22,29), (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([(1,9,30,17),(2,22,31,14),(3,11,32,19),(4,24,25,16),(5,13,26,21),(6,18,27,10),(7,15,28,23),(8,20,29,12)], [(1,7,5,3),(2,8,6,4),(9,15,13,11),(10,16,14,12),(17,23,21,19),(18,24,22,20),(25,31,29,27),(26,32,30,28)], [(1,17,26,13),(2,24),(3,11,28,23),(4,10),(5,21,30,9),(6,20),(7,15,32,19),(8,14),(12,27),(16,31),(18,25),(22,29)], [(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 2F 2G 4A ··· 4H 4I 4J 4K ··· 4P 8A ··· 8H order 1 2 2 2 2 2 2 2 4 ··· 4 4 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 2 8 8 2 ··· 2 4 4 8 ··· 8 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 D4 D4 D8 SD16 C23.C23 C42⋊C22 kernel C42.61D4 C22.SD16 C42⋊8C4 C42.12C4 C22.26C24 C4×D4 C4⋊1D4 C4⋊Q8 C42 C22×C4 C2×C4 C2×C4 C2 C2 # reps 1 4 1 1 1 4 2 2 2 2 4 4 2 2

Matrix representation of C42.61D4 in GL6(𝔽17)

 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 2 0 1 0 0 15 0 1 0
,
 0 16 0 0 0 0 1 0 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
,
 0 4 0 0 0 0 4 0 0 0 0 0 0 0 0 16 0 0 0 0 1 0 0 0 0 0 0 12 0 13 0 0 12 0 4 0
,
 5 12 0 0 0 0 5 5 0 0 0 0 0 0 0 9 0 9 0 0 9 0 8 0 0 0 0 0 0 9 0 0 0 0 9 0

`G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,15,0,0,1,0,2,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,12,0,0,16,0,12,0,0,0,0,0,0,4,0,0,0,0,13,0],[5,5,0,0,0,0,12,5,0,0,0,0,0,0,0,9,0,0,0,0,9,0,0,0,0,0,0,8,0,9,0,0,9,0,9,0] >;`

C42.61D4 in GAP, Magma, Sage, TeX

`C_4^2._{61}D_4`
`% in TeX`

`G:=Group("C4^2.61D4");`
`// GroupNames label`

`G:=SmallGroup(128,249);`
`// by ID`

`G=gap.SmallGroup(128,249);`
`# by ID`

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

`G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^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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