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G = C23≀C2  order 128 = 27

Wreath product of C23 by C2

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

Aliases: C23≀C2, C26⋊1C2, C24⋊15D4, C24⋊1C23, C25.88C22, C23.746C24, C22⋊3C22≀C2, (C22×C4)⋊1C23, C24⋊3C4⋊28C2, C23.628(C2×D4), (C22×D4)⋊15C22, C22.456(C22×D4), (C2×C22≀C2)⋊17C2, C2.29(C2×C22≀C2), (C2×C22⋊C4)⋊33C22, 2-Sylow(SO+(4,8)), SmallGroup(128,1578)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C23 — C23≀C2
Chief series
C1 — C2 — C22 — C23 — C24 — C25 — C26 — C23≀C2
Lower central
C1 — C23 — C23≀C2
Upper central
C1 — C23 — C23≀C2
Jennings
C1 — C23 — C23≀C2

Subgroups: 3332 in 1711 conjugacy classes, 180 normal (5 characteristic)
C1, C2 [×7], C2 [×29], C4 [×7], C22 [×35], C22 [×315], C2×C4 [×21], D4 [×28], C23, C23 [×42], C23 [×683], C22⋊C4 [×42], C22×C4 [×7], C2×D4 [×42], C24, C24 [×35], C24 [×308], C2×C22⋊C4 [×21], C22≀C2 [×28], C22×D4 [×7], C25 [×7], C25 [×28], C24⋊3C4 [×7], C2×C22≀C2 [×7], C26, C23≀C2

Quotients:
C1, C2 [×15], C22 [×35], D4 [×28], C23 [×15], C2×D4 [×42], C24, C22≀C2 [×28], C22×D4 [×7], C2×C22≀C2 [×7], C23≀C2

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=g2=1, ab=ba, ac=ca, gag=ad=da, ae=ea, af=fa, bc=cb, bd=db, gbg=be=eb, bf=fb, cd=dc, ce=ec, gcg=cf=fc, de=ed, df=fd, dg=gd, ef=fe, eg=ge, fg=gf >

Permutation representations
►On 16 points - transitive group 16T325
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)

(1 15)(2 16)(3 4)(5 6)(7 10)(8 9)(11 12)(13 14)

(1 8)(2 7)(3 6)(4 5)(9 15)(10 16)(11 13)(12 14)

(1 7)(2 8)(3 11)(4 12)(5 14)(6 13)(9 16)(10 15)

(1 9)(2 10)(3 13)(4 14)(5 12)(6 11)(7 16)(8 15)

(1 10)(2 9)(3 5)(4 6)(7 15)(8 16)(11 14)(12 13)

(1 6)(2 14)(3 16)(4 10)(5 8)(7 13)(9 11)(12 15)
 
G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (1,15)(2,16)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14), (1,8)(2,7)(3,6)(4,5)(9,15)(10,16)(11,13)(12,14), (1,7)(2,8)(3,11)(4,12)(5,14)(6,13)(9,16)(10,15), (1,9)(2,10)(3,13)(4,14)(5,12)(6,11)(7,16)(8,15), (1,10)(2,9)(3,5)(4,6)(7,15)(8,16)(11,14)(12,13), (1,6)(2,14)(3,16)(4,10)(5,8)(7,13)(9,11)(12,15)>;
 
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (1,15)(2,16)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14), (1,8)(2,7)(3,6)(4,5)(9,15)(10,16)(11,13)(12,14), (1,7)(2,8)(3,11)(4,12)(5,14)(6,13)(9,16)(10,15), (1,9)(2,10)(3,13)(4,14)(5,12)(6,11)(7,16)(8,15), (1,10)(2,9)(3,5)(4,6)(7,15)(8,16)(11,14)(12,13), (1,6)(2,14)(3,16)(4,10)(5,8)(7,13)(9,11)(12,15) );
 
G=PermutationGroup([(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)], [(1,15),(2,16),(3,4),(5,6),(7,10),(8,9),(11,12),(13,14)], [(1,8),(2,7),(3,6),(4,5),(9,15),(10,16),(11,13),(12,14)], [(1,7),(2,8),(3,11),(4,12),(5,14),(6,13),(9,16),(10,15)], [(1,9),(2,10),(3,13),(4,14),(5,12),(6,11),(7,16),(8,15)], [(1,10),(2,9),(3,5),(4,6),(7,15),(8,16),(11,14),(12,13)], [(1,6),(2,14),(3,16),(4,10),(5,8),(7,13),(9,11),(12,15)])
 
G:=TransitiveGroup(16,325);
 

Matrix representation ►G ⊆ GL6(ℤ)

-100000
0-10000
00-1000
000100
0000-10
000001
,
-100000
110000
001000
000-100
000010
000001
,
-100000
0-10000
001000
000100
0000-10
000001
,
100000
010000
00-1000
000-100
0000-10
00000-1
,
-100000
0-10000
00-1000
000-100
000010
000001
,
100000
010000
001000
000100
0000-10
00000-1
,
-1-20000
010000
000100
001000
000001
000010

G:=sub<GL(6,Integers())| [-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,-1,0,0,0,0,0,0,1],[-1,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-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,-1,0,0,0,0,0,0,1],[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,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
 

44 conjugacy classes

class 1 2A···2G2H···2AI2AJ4A···4G
order12···22···224···4
size11···12···288···8

44 irreducible representations

dim11112
type+++++
imageC1C2C2C2D4
kernelC23≀C2C24⋊3C4C2×C22≀C2C26C24
# reps177128

In GAP, Magma, Sage, TeX 

C_2^3\wr C_2
 
% in TeX
 
G:=Group("C2^3wrC2");
 
// GroupNames label
 
G:=SmallGroup(128,1578);
 
// by ID
 
G=gap.SmallGroup(128,1578);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,758,2019]);
 
// 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,g*a*g=a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,g*b*g=b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,g*c*g=c*f=f*c,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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