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## G = C24.63D4order 128 = 27

### 18th non-split extension by C24 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 — C4 — C24.63D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C42⋊C2 — C24.63D4
 Lower central C1 — C2 — C4 — C24.63D4
 Upper central C1 — C2×C4 — C23×C4 — C24.63D4
 Jennings C1 — C2 — C2 — C22×C4 — C24.63D4

Generators and relations for C24.63D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=b, ab=ba, ac=ca, eae-1=ad=da, af=fa, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf-1=cd=dc, de=ed, df=fd, fef-1=bcde3 >

Subgroups: 340 in 202 conjugacy classes, 108 normal (32 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×4], C4 [×4], C4 [×8], C22 [×3], C22 [×4], C22 [×10], C8 [×4], C2×C4 [×8], C2×C4 [×20], C2×C4 [×16], C23 [×3], C23 [×4], C23 [×2], C42 [×4], C42 [×6], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×6], M4(2) [×4], M4(2) [×6], C22×C4 [×6], C22×C4 [×8], C22×C4 [×4], C24, C2×C42 [×2], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4, C42⋊C2 [×8], C42⋊C2 [×4], C22×C8, C2×M4(2) [×6], C2×M4(2) [×3], C23×C4, C426C4 [×4], C2×C42⋊C2 [×2], C22×M4(2), C24.63D4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×6], Q8 [×2], C23, C42 [×4], C22⋊C4 [×12], C4⋊C4 [×12], C22×C4 [×3], C2×D4 [×3], C2×Q8, C2.C42 [×8], C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4 [×3], C2×C2.C42, C42⋊C22 [×2], C24.63D4

Smallest permutation representation of C24.63D4
On 32 points
Generators in S32
```(1 19)(2 24)(3 21)(4 18)(5 23)(6 20)(7 17)(8 22)(9 26)(10 31)(11 28)(12 25)(13 30)(14 27)(15 32)(16 29)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 17)
(1 5)(3 7)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 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)
(1 9 31 18)(2 17 32 16)(3 11 25 20)(4 19 26 10)(5 13 27 22)(6 21 28 12)(7 15 29 24)(8 23 30 14)```

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 4 type + + + + + - + image C1 C2 C2 C2 C4 C4 C4 D4 Q8 D4 C42⋊C22 kernel C24.63D4 C42⋊6C4 C2×C42⋊C2 C22×M4(2) C2×C4⋊C4 C42⋊C2 C2×M4(2) C22×C4 C22×C4 C24 C2 # reps 1 4 2 1 4 12 8 5 2 1 4

Matrix representation of C24.63D4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 15 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 16 1 0
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 1 0 1 0 0 0 16 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 4 1 0 0 0 0 2 13 0 0 0 0 0 0 16 0 0 2 0 0 1 0 1 16 0 0 7 13 0 16 0 0 6 0 0 1
,
 13 16 0 0 0 0 0 4 0 0 0 0 0 0 16 0 15 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 16 0

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

C24.63D4 in GAP, Magma, Sage, TeX

`C_2^4._{63}D_4`
`% in TeX`

`G:=Group("C2^4.63D4");`
`// GroupNames label`

`G:=SmallGroup(128,465);`
`// by ID`

`G=gap.SmallGroup(128,465);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,723,2019,248,4037]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d,f^2=b,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,e*c*e^-1=f*c*f^-1=c*d=d*c,d*e=e*d,d*f=f*d,f*e*f^-1=b*c*d*e^3>;`
`// generators/relations`

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