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## G = C24.90D4order 128 = 27

### 45th non-split extension by C24 of D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C24.90D4
 Chief series C1 — C2 — C22 — C23 — C24 — C25 — D4×C23 — C24.90D4
 Lower central C1 — C22 — C24.90D4
 Upper central C1 — C23 — C24.90D4
 Jennings C1 — C23 — C24.90D4

Generators and relations for C24.90D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=d, ab=ba, faf-1=ac=ca, ad=da, ae=ea, ebe-1=fbf-1=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=de-1 >

Subgroups: 1260 in 580 conjugacy classes, 180 normal (10 characteristic)
C1, C2, C2 [×6], C2 [×16], C4 [×12], C22, C22 [×18], C22 [×88], C2×C4 [×4], C2×C4 [×52], D4 [×32], C23, C23 [×38], C23 [×80], C22⋊C4 [×24], C22×C4 [×14], C22×C4 [×20], C2×D4 [×16], C2×D4 [×48], C24, C24 [×20], C24 [×12], C2.C42 [×8], C2×C22⋊C4 [×16], C2×C22⋊C4 [×8], C23×C4, C23×C4 [×4], C22×D4 [×12], C22×D4 [×8], C25 [×2], C243C4 [×2], C23.34D4 [×2], C23.23D4 [×8], C22×C22⋊C4 [×2], D4×C23, C24.90D4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C24, C2×C22⋊C4 [×12], C23×C4, C22×D4 [×2], 2+ 1+4 [×4], C22×C22⋊C4, C22.11C24 [×2], C233D4 [×4], C24.90D4

Smallest permutation representation of C24.90D4
On 32 points
Generators in S32
```(1 22)(2 23)(3 24)(4 21)(5 13)(6 14)(7 15)(8 16)(9 30)(10 31)(11 32)(12 29)(17 26)(18 27)(19 28)(20 25)
(1 3)(2 8)(4 6)(5 7)(9 28)(10 12)(11 26)(13 15)(14 21)(16 23)(17 32)(18 20)(19 30)(22 24)(25 27)(29 31)
(1 5)(2 6)(3 7)(4 8)(9 26)(10 27)(11 28)(12 25)(13 22)(14 23)(15 24)(16 21)(17 30)(18 31)(19 32)(20 29)
(1 27)(2 28)(3 25)(4 26)(5 10)(6 11)(7 12)(8 9)(13 31)(14 32)(15 29)(16 30)(17 21)(18 22)(19 23)(20 24)
(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 4 27 26)(2 25 28 3)(5 8 10 9)(6 12 11 7)(13 21 31 17)(14 20 32 24)(15 23 29 19)(16 18 30 22)```

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2S 2T 2U 2V 2W 4A ··· 4T order 1 2 ··· 2 2 ··· 2 2 2 2 2 4 ··· 4 size 1 1 ··· 1 2 ··· 2 4 4 4 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 4 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 D4 2+ 1+4 kernel C24.90D4 C24⋊3C4 C23.34D4 C23.23D4 C22×C22⋊C4 D4×C23 C22×D4 C24 C22 # reps 1 2 2 8 2 1 16 8 4

Matrix representation of C24.90D4 in GL8(𝔽5)

 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 3 0 0 0 0 0 0 0 1
,
 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4
,
 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 3 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 4 4 0 0 0 0 4 0 0 0 0 0 0 0 1 1 0 0
,
 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0

`G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,3,1],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,0,1,0,0,0,0,1,4,0,0,0,0,0,0,0,4,0,0],[0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;`

C24.90D4 in GAP, Magma, Sage, TeX

`C_2^4._{90}D_4`
`% in TeX`

`G:=Group("C2^4.90D4");`
`// GroupNames label`

`G:=SmallGroup(128,1040);`
`// by ID`

`G=gap.SmallGroup(128,1040);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,219,675]);`
`// Polycyclic`

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

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