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## 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, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C24, C2×C42, C2×C22⋊C4, C22≀C2, C4.4D4, C22×D4, C22×Q8, C25, C2×C22≀C2, C2×C4.4D4, C24⋊C22, C2×C24⋊C22
Quotients: C1, C2, C22, C23, C24, 2+ 1+4, C25, C24⋊C22, C2×2+ 1+4, 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 11)(4 12)(7 8)(9 16)(10 15)(13 20)(14 19)(17 27)(18 28)(21 23)(22 24)(25 26)(29 30)(31 32)
(1 4)(2 3)(5 11)(6 12)(7 32)(8 31)(9 16)(10 15)(13 20)(14 19)(17 18)(21 22)(23 24)(25 29)(26 30)(27 28)
(1 5)(2 6)(3 12)(4 11)(7 31)(8 32)(9 14)(10 13)(15 20)(16 19)(17 22)(18 21)(23 28)(24 27)(25 30)(26 29)
(1 12)(2 11)(3 5)(4 6)(7 30)(8 29)(9 15)(10 16)(13 19)(14 20)(17 23)(18 24)(21 27)(22 28)(25 31)(26 32)
(1 18)(2 17)(3 27)(4 28)(5 21)(6 22)(7 10)(8 9)(11 23)(12 24)(13 31)(14 32)(15 29)(16 30)(19 25)(20 26)
(1 10)(2 9)(3 19)(4 20)(5 13)(6 14)(7 18)(8 17)(11 15)(12 16)(21 31)(22 32)(23 29)(24 30)(25 27)(26 28)

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

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

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

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

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