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## G = C24.94D4order 128 = 27

### 49th 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 — C23 — C24.94D4
 Chief series C1 — C2 — C22 — C23 — C24 — C23×C4 — C22×C22⋊C4 — C24.94D4
 Lower central C1 — C23 — C24.94D4
 Upper central C1 — C23 — C24.94D4
 Jennings C1 — C23 — C24.94D4

Generators and relations for C24.94D4
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=ce-1 >

Subgroups: 1236 in 558 conjugacy classes, 124 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×14], C4 [×12], C22, C22 [×14], C22 [×86], C2×C4 [×4], C2×C4 [×44], D4 [×32], C23, C23 [×18], C23 [×94], C22⋊C4 [×30], C4⋊C4 [×8], C22×C4 [×10], C22×C4 [×16], C2×D4 [×8], C2×D4 [×52], C24 [×3], C24 [×6], C24 [×22], C2.C42 [×4], C2×C22⋊C4 [×18], C2×C22⋊C4 [×4], C2×C4⋊C4 [×4], C22.D4 [×8], C23×C4 [×3], C22×D4 [×4], C22×D4 [×12], C25 [×2], C243C4, C243C4 [×2], C23.8Q8 [×2], C23.23D4 [×2], C23.10D4 [×4], C22×C22⋊C4, C2×C22.D4 [×2], D4×C23, C24.94D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C4○D4 [×4], C24, C22≀C2 [×4], C22.D4 [×4], C22×D4 [×3], C2×C4○D4 [×2], 2+ 1+4 [×2], C2×C22≀C2, C2×C22.D4, C233D4, D45D4 [×4], C24.94D4

Smallest permutation representation of C24.94D4
On 32 points
Generators in S32
```(1 19)(2 29)(3 17)(4 31)(5 7)(6 12)(8 10)(9 11)(13 23)(14 16)(15 21)(18 27)(20 25)(22 24)(26 30)(28 32)
(1 26)(2 29)(3 28)(4 31)(5 16)(6 12)(7 14)(8 10)(9 24)(11 22)(13 23)(15 21)(17 32)(18 27)(19 30)(20 25)
(1 17)(2 18)(3 19)(4 20)(5 22)(6 23)(7 24)(8 21)(9 14)(10 15)(11 16)(12 13)(25 31)(26 32)(27 29)(28 30)
(1 30)(2 31)(3 32)(4 29)(5 14)(6 15)(7 16)(8 13)(9 22)(10 23)(11 24)(12 21)(17 28)(18 25)(19 26)(20 27)
(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 14)(2 12)(3 16)(4 10)(5 30)(6 27)(7 32)(8 25)(9 17)(11 19)(13 18)(15 20)(21 31)(22 28)(23 29)(24 26)```

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 2P ··· 2U 4A ··· 4L 4M 4N 4O 4P 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 1 1 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 C4○D4 2+ 1+4 kernel C24.94D4 C24⋊3C4 C23.8Q8 C23.23D4 C23.10D4 C22×C22⋊C4 C2×C22.D4 D4×C23 C2×D4 C24 C23 C22 # reps 1 3 2 2 4 1 2 1 8 4 8 2

Matrix representation of C24.94D4 in GL6(𝔽5)

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

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

C24.94D4 in GAP, Magma, Sage, TeX

`C_2^4._{94}D_4`
`% in TeX`

`G:=Group("C2^4.94D4");`
`// GroupNames label`

`G:=SmallGroup(128,1137);`
`// by ID`

`G=gap.SmallGroup(128,1137);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,2,253,758,723,100,675]);`
`// 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=c*e^-1>;`
`// generators/relations`

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