Copied to
clipboard

## G = C2×C24⋊C22order 128 = 27

### Direct product of C2 and C24⋊C22

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C2×C24⋊C22
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, fbf=be=eb, gbg=bde, gcg=cd=dc, ce=ec, fcf=cde, de=ed, df=fd, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 1436 in 704 conjugacy classes, 388 normal (4 characteristic)
C1, C2 [×7], C2 [×12], C4 [×18], C22, C22 [×6], C22 [×92], C2×C4 [×18], C2×C4 [×18], D4 [×36], Q8 [×12], C23, C23 [×12], C23 [×92], C42 [×12], C22⋊C4 [×72], C22×C4 [×9], C2×D4 [×36], C2×D4 [×18], C2×Q8 [×12], C2×Q8 [×6], C24 [×14], C24 [×12], C2×C42 [×3], C2×C22⋊C4 [×18], C22≀C2 [×48], C4.4D4 [×72], C22×D4 [×9], C22×Q8 [×3], C25 [×2], C2×C22≀C2 [×6], C2×C4.4D4 [×9], C24⋊C22 [×16], C2×C24⋊C22
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], C24 [×31], 2+ 1+4 [×6], C25, C24⋊C22 [×4], C2×2+ 1+4 [×3], C2×C24⋊C22

Smallest permutation representation of C2×C24⋊C22
On 32 points
Generators in S32
(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 6)(2 5)(3 15)(4 16)(7 11)(8 12)(9 22)(10 21)(13 14)(17 18)(19 30)(20 29)(23 28)(24 27)(25 26)(31 32)
(1 16)(2 15)(3 5)(4 6)(7 11)(8 12)(9 10)(13 25)(14 26)(17 31)(18 32)(19 20)(21 22)(23 28)(24 27)(29 30)
(1 5)(2 6)(3 16)(4 15)(7 23)(8 24)(9 20)(10 19)(11 28)(12 27)(13 26)(14 25)(17 32)(18 31)(21 30)(22 29)
(1 4)(2 3)(5 15)(6 16)(7 12)(8 11)(9 30)(10 29)(13 31)(14 32)(17 25)(18 26)(19 22)(20 21)(23 27)(24 28)
(1 29)(2 30)(3 9)(4 10)(5 22)(6 21)(7 32)(8 31)(11 13)(12 14)(15 19)(16 20)(17 23)(18 24)(25 27)(26 28)
(1 24)(2 23)(3 27)(4 28)(5 8)(6 7)(9 25)(10 26)(11 15)(12 16)(13 19)(14 20)(17 30)(18 29)(21 32)(22 31)

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2S 4A ··· 4R order 1 2 ··· 2 2 ··· 2 4 ··· 4 size 1 1 ··· 1 4 ··· 4 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 4 type + + + + + image C1 C2 C2 C2 2+ 1+4 kernel C2×C24⋊C22 C2×C22≀C2 C2×C4.4D4 C24⋊C22 C22 # reps 1 6 9 16 6

Matrix representation of C2×C24⋊C22 in GL12(ℤ)

 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 -1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1
,
 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 1 0 0 0 0 0 0 0 0 0 -1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -1
,
 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1
,
 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1
,
 -1 0 2 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -2 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 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 0 0 0 0 0 0 0 0 1 0 0
,
 -1 2 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 -1 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -2 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 -1 0 1 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0

G:=sub<GL(12,Integers())| [1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1],[1,1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1],[-1,0,-1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,2,1,1,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[-1,0,0,0,0,0,0,0,0,0,0,0,2,1,1,1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,-2,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0] >;

C2×C24⋊C22 in GAP, Magma, Sage, TeX

C_2\times C_2^4\rtimes C_2^2
% in TeX

G:=Group("C2xC2^4:C2^2");
// GroupNames label

G:=SmallGroup(128,2258);
// by ID

G=gap.SmallGroup(128,2258);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,232,1430,1059,2915,570]);
// Polycyclic

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

׿
×
𝔽