C2^2xC2^2wrC2 - GroupNames

G = C₂²xC₂²wrC₂ order 128 = 2⁷

Direct product of C₂² and C₂²wrC₂

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

Aliases: C₂²xC₂²wrC₂, C₂⁶:₂C₂, C₂⁴:₁₆D₄, C₂³:₁C₂⁴, C₂⁴:₁₁C₂³, C₂⁵:₁₀C₂², C₂².₂₀C₂⁵, (C₂xC₄):₁C₂⁴, (D₄xC₂³):₉C₂, C₂³:₁₀(C₂xD₄), C₂.₄(D₄xC₂³), (C₂xD₄):₁₃C₂³, C₂²:₃(C₂²xD₄), C₂²:C₄:₁₄C₂³, (C₂²xC₄):₁₁C₂³, (C₂³xC₄):₂₇C₂², (C₂²xD₄):₅₆C₂², (C₂²xC₂²:C₄):₂₆C₂, (C₂xC₂²:C₄):₈₀C₂², SmallGroup(128,2163)

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

Derived series

C₁ — C₂² — C₂²xC₂²wrC₂

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂⁵ — C₂⁶ — C₂²xC₂²wrC₂

Lower central

C₁ — C₂² — C₂²xC₂²wrC₂

Upper central

C₁ — C₂⁴ — C₂²xC₂²wrC₂

Jennings

C₁ — C₂² — C₂²xC₂²wrC₂

Generators and relations for C₂²xC₂²wrC₂
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, be=eb, bf=fb, bg=gb, cd=dc, gcg=ce=ec, cf=fc, de=ed, gdg=df=fd, ef=fe, eg=ge, fg=gf >

Subgroups: 4076 in 2272 conjugacy classes, 556 normal (6 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂², C₂xC₄, C₂xC₄, D₄, C₂³, C₂³, C₂²:C₄, C₂²xC₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂⁴, C₂⁴, C₂⁴, C₂xC₂²:C₄, C₂²wrC₂, C₂³xC₄, C₂²xD₄, C₂²xD₄, C₂⁵, C₂⁵, C₂⁵, C₂²xC₂²:C₄, C₂xC₂²wrC₂, D₄xC₂³, C₂⁶, C₂²xC₂²wrC₂
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂xD₄, C₂⁴, C₂²wrC₂, C₂²xD₄, C₂⁵, C₂xC₂²wrC₂, D₄xC₂³, C₂²xC₂²wrC₂

Smallest permutation representation of C₂²xC₂²wrC₂
►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 4)(2 3)(5 7)(6 8)(9 23)(10 24)(11 13)(12 14)(15 18)(16 17)(19 21)(20 22)(25 27)(26 28)(29 31)(30 32)

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

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

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

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

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

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

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

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

56 conjugacy classes

class	1	2A	···	2O	2P	···	2AM	2AN	2AO	2AP	2AQ	4A	···	4L
order	1	2	···	2	2	···	2	2	2	2	2	4	···	4
size	1	1	···	1	2	···	2	4	4	4	4	4	···	4

56 irreducible representations

dim	1	1	1	1	1	2
type	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	D₄
kernel	C₂²xC₂²wrC₂	C₂²xC₂²:C₄	C₂xC₂²wrC₂	D₄xC₂³	C₂⁶	C₂⁴
# reps	1	3	24	3	1	24

Matrix representation of C₂²xC₂²wrC₂ ►in GL₆(Z)

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

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],[-1,0,0,0,0,0,0,-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,-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,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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,-1,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,-2,1] >;

C₂²xC₂²wrC₂ in GAP, Magma, Sage, TeX

C_2^2\times C_2^2\wr C_2

% in TeX

G:=Group("C2^2xC2^2wrC2");

// GroupNames label

G:=SmallGroup(128,2163);

// by ID

G=gap.SmallGroup(128,2163);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430]);

// 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,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,g*c*g=c*e=e*c,c*f=f*c,d*e=e*d,g*d*g=d*f=f*d,e*f=f*e,e*g=g*e,f*g=g*f>;

// generators/relations

G = C22xC22wrC2 order 128 = 27

Direct product of C22 and C22wrC2

G = C₂²xC₂²wrC₂ order 128 = 2⁷

Direct product of C₂² and C₂²wrC₂