Copied to
clipboard

## G = C23.4C24order 128 = 27

### 4th non-split extension by C23 of C24 acting via C24/C22=C22

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C23.4C24
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C22×Q8 — C2×2- 1+4 — C23.4C24
 Lower central C1 — C2 — C22 — C23.4C24
 Upper central C1 — C2 — C22×Q8 — C23.4C24
 Jennings C1 — C2 — C23 — C23.4C24

Generators and relations for C23.4C24
G = < a,b,c,d,e,f,g | a2=b2=c2=e2=1, d2=b, f2=g2=c, ab=ba, dad-1=ac=ca, ae=ea, af=fa, ag=ga, ebe=bc=cb, bd=db, bf=fb, bg=gb, cd=dc, ce=ec, gfg-1=cf=fc, cg=gc, ede=acd, df=fd, dg=gd, ef=fe, eg=ge >

Subgroups: 580 in 358 conjugacy classes, 170 normal (7 characteristic)
C1, C2, C2 [×7], C4 [×12], C4 [×12], C22, C22 [×2], C22 [×9], C2×C4 [×22], C2×C4 [×32], D4 [×20], Q8 [×16], Q8 [×12], C23, C23 [×4], C42 [×12], C22⋊C4 [×8], C4⋊C4 [×12], C22×C4 [×15], C2×D4 [×10], C2×Q8 [×18], C2×Q8 [×16], C4○D4 [×40], C23⋊C4 [×16], C42⋊C2 [×12], C4×Q8 [×8], C22×Q8, C22×Q8 [×4], C2×C4○D4 [×10], 2- 1+4 [×8], C23.C23 [×12], C23.32C23 [×2], C2×2- 1+4, C23.4C24
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], C22×C22⋊C4, C23.4C24

Smallest permutation representation of C23.4C24
On 32 points
Generators in S32
```(2 28)(4 26)(6 17)(8 19)(10 30)(12 32)(14 22)(16 24)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)
(1 27)(2 28)(3 25)(4 26)(5 20)(6 17)(7 18)(8 19)(9 29)(10 30)(11 31)(12 32)(13 21)(14 22)(15 23)(16 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 5)(2 17)(3 18)(4 8)(6 28)(7 25)(9 23)(10 16)(11 13)(12 22)(14 32)(15 29)(19 26)(20 27)(21 31)(24 30)
(1 31 27 11)(2 32 28 12)(3 29 25 9)(4 30 26 10)(5 21 20 13)(6 22 17 14)(7 23 18 15)(8 24 19 16)
(1 15 27 23)(2 16 28 24)(3 13 25 21)(4 14 26 22)(5 29 20 9)(6 30 17 10)(7 31 18 11)(8 32 19 12)```

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

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

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

41 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A ··· 4L 4M ··· 4AF order 1 2 2 2 2 2 2 2 2 4 ··· 4 4 ··· 4 size 1 1 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4

41 irreducible representations

 dim 1 1 1 1 1 1 2 8 type + + + + + - image C1 C2 C2 C2 C4 C4 D4 C23.4C24 kernel C23.4C24 C23.C23 C23.32C23 C2×2- 1+4 C22×Q8 C2×C4○D4 C2×Q8 C1 # reps 1 12 2 1 8 8 8 1

Matrix representation of C23.4C24 in GL8(𝔽5)

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

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

C23.4C24 in GAP, Magma, Sage, TeX

`C_2^3._4C_2^4`
`% in TeX`

`G:=Group("C2^3.4C2^4");`
`// GroupNames label`

`G:=SmallGroup(128,1616);`
`// by ID`

`G=gap.SmallGroup(128,1616);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,352,521,248,2804,2028]);`
`// Polycyclic`

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

׿
×
𝔽