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G = C24⋊7D4order 128 = 27

2nd semidirect product of C24 and D4 acting via D4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C24⋊7D4
 Chief series C1 — C2 — C22 — C23 — C24 — C25 — D4×C23 — C24⋊7D4
 Lower central C1 — C23 — C24⋊7D4
 Upper central C1 — C23 — C24⋊7D4
 Jennings C1 — C23 — C24⋊7D4

Generators and relations for C247D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=f2=1, ab=ba, faf=ac=ca, ad=da, eae-1=acd, ebe-1=fbf=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 1780 in 782 conjugacy classes, 140 normal (8 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C22⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C24, C2.C42, C2×C22⋊C4, C22≀C2, C23×C4, C22×D4, C22×D4, C25, C25, C243C4, C23.23D4, C232D4, C2×C22≀C2, D4×C23, C247D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22≀C2, C22×D4, 2+ 1+4, C2×C22≀C2, C233D4, D42, C247D4

Smallest permutation representation of C247D4
On 32 points
Generators in S32
```(1 29)(2 12)(3 31)(4 10)(5 18)(6 23)(7 20)(8 21)(9 16)(11 14)(13 32)(15 30)(17 28)(19 26)(22 25)(24 27)
(1 27)(2 13)(3 25)(4 15)(5 16)(6 28)(7 14)(8 26)(9 18)(10 30)(11 20)(12 32)(17 23)(19 21)(22 31)(24 29)
(1 5)(2 6)(3 7)(4 8)(9 24)(10 21)(11 22)(12 23)(13 28)(14 25)(15 26)(16 27)(17 32)(18 29)(19 30)(20 31)
(1 25)(2 26)(3 27)(4 28)(5 14)(6 15)(7 16)(8 13)(9 20)(10 17)(11 18)(12 19)(21 32)(22 29)(23 30)(24 31)
(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 2)(3 4)(5 6)(7 8)(9 17)(10 20)(11 19)(12 18)(13 16)(14 15)(21 31)(22 30)(23 29)(24 32)(25 26)(27 28)```

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 2P ··· 2Y 4A ··· 4H 4I 4J 4K 4L order 1 2 ··· 2 2 ··· 2 2 ··· 2 4 ··· 4 4 4 4 4 size 1 1 ··· 1 2 ··· 2 4 ··· 4 4 ··· 4 8 8 8 8

38 irreducible representations

 dim 1 1 1 1 1 1 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 2+ 1+4 kernel C24⋊7D4 C24⋊3C4 C23.23D4 C23⋊2D4 C2×C22≀C2 D4×C23 C2×D4 C24 C22 # reps 1 1 4 4 4 2 16 4 2

Matrix representation of C247D4 in GL6(𝔽5)

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

`G:=sub<GL(6,GF(5))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,2,2,0,0,0,0,0,0,0,1,0,0,0,0,4,0],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,2,4,0,0,0,0,3,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;`

C247D4 in GAP, Magma, Sage, TeX

`C_2^4\rtimes_7D_4`
`% in TeX`

`G:=Group("C2^4:7D4");`
`// GroupNames label`

`G:=SmallGroup(128,1135);`
`// by ID`

`G=gap.SmallGroup(128,1135);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,2,253,758,723,675]);`
`// Polycyclic`

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

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