C2xC2wrC2^2 - GroupNames

G = C₂×C₂≀C₂² order 128 = 2⁷

Direct product of C₂ and C₂≀C₂²

direct product, p-group, metabelian, nilpotent (class 3), monomial, rational

Aliases: C₂×C₂≀C₂², C₂⁴⋊₄D₄, C₂⁴⋊₆C₂³, C₂⁵⋊₃C₂², C₂³.₅C₂⁴, 2₊¹⁺⁴⋊₇C₂², C₂³⋊(C₂×D₄), (C₂×D₄)⋊₂₄D₄, (C₂²×C₄)⋊₅D₄, C₂³⋊C₄⋊₅C₂², C₂²⋊C₄⋊₁C₂³, (C₂×D₄).₃₉C₂³, C₂²≀C₂⋊₂₃C₂², C₂².₂₅C₂²≀C₂, (C₂×2₊¹⁺⁴)⋊₅C₂, C₂².₃₉(C₂²×D₄), (C₂²×D₄).₃₃₂C₂², (C₂×C₄)⋊(C₂×D₄), (C₂×C₂³⋊C₄)⋊₁₆C₂, (C₂×C₂²≀C₂)⋊₁₉C₂, C₂.₆₀(C₂×C₂²≀C₂), (C₂×C₂²⋊C₄)⋊₃₆C₂², SmallGroup(128,1755)

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

Derived series

C₁ — C₂³ — C₂×C₂≀C₂²

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂²×D₄ — C₂×2₊¹⁺⁴ — C₂×C₂≀C₂²

Lower central

C₁ — C₂ — C₂³ — C₂×C₂≀C₂²

Upper central

C₁ — C₂² — C₂⁴ — C₂×C₂≀C₂²

Jennings

C₁ — C₂ — C₂³ — C₂×C₂≀C₂²

Generators and relations for C₂×C₂≀C₂²
G = < a,b,c,d,e,f | a²=b²=c²=d²=e⁴=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe^-1=fbf=bcd, ece^-1=cd=dc, cf=fc, de=ed, df=fd, fef=e^-1 >

Subgroups: 1236 in 509 conjugacy classes, 108 normal (8 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₂²⋊C₄, C₂²⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₂⁴, C₂⁴, C₂³⋊C₄, C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂²≀C₂, C₂²≀C₂, C₂²×D₄, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2₊¹⁺⁴, C₂⁵, C₂×C₂³⋊C₄, C₂≀C₂², C₂×C₂²≀C₂, C₂×2₊¹⁺⁴, C₂×C₂≀C₂²
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂⁴, C₂²≀C₂, C₂²×D₄, C₂≀C₂², C₂×C₂²≀C₂, C₂×C₂≀C₂²

Permutation representations of C₂×C₂≀C₂²
►On 16 points - transitive group 16T245

Generators in S₁₆

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

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

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

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

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

(1 3)(2 4)(5 8)(6 7)(9 11)(14 16)

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

G:=Group( (1,6)(2,5)(3,7)(4,8)(9,14)(10,15)(11,16)(12,13), (1,11)(2,13)(3,14)(4,10)(5,12)(6,16)(7,9)(8,15), (1,3)(2,5)(4,8)(6,7)(9,16)(10,15)(11,14)(12,13), (1,7)(2,8)(3,6)(4,5)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,3)(2,4)(5,8)(6,7)(9,11)(14,16) );

G=PermutationGroup([[(1,6),(2,5),(3,7),(4,8),(9,14),(10,15),(11,16),(12,13)], [(1,11),(2,13),(3,14),(4,10),(5,12),(6,16),(7,9),(8,15)], [(1,3),(2,5),(4,8),(6,7),(9,16),(10,15),(11,14),(12,13)], [(1,7),(2,8),(3,6),(4,5),(9,11),(10,12),(13,15),(14,16)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,3),(2,4),(5,8),(6,7),(9,11),(14,16)]])

G:=TransitiveGroup(16,245);

►On 16 points - transitive group 16T246

Generators in S₁₆

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

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

(2 7)(3 6)(10 12)(13 15)

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

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

(9 11)(14 16)

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

G:=Group( (1,5)(2,6)(3,7)(4,8)(9,16)(10,13)(11,14)(12,15), (1,11)(2,10)(3,15)(4,16)(5,14)(6,13)(7,12)(8,9), (2,7)(3,6)(10,12)(13,15), (1,8)(2,7)(3,6)(4,5)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (9,11)(14,16) );

G=PermutationGroup([[(1,5),(2,6),(3,7),(4,8),(9,16),(10,13),(11,14),(12,15)], [(1,11),(2,10),(3,15),(4,16),(5,14),(6,13),(7,12),(8,9)], [(2,7),(3,6),(10,12),(13,15)], [(1,8),(2,7),(3,6),(4,5),(9,11),(10,12),(13,15),(14,16)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(9,11),(14,16)]])

G:=TransitiveGroup(16,246);

►On 16 points - transitive group 16T271

Generators in S₁₆

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

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

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

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

(5 6)(7 8)(9 10 11 12)(13 14 15 16)

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

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

G:=Group( (1,4)(2,3)(5,8)(6,7)(9,15)(10,16)(11,13)(12,14), (1,16)(2,14)(3,12)(4,10)(5,13)(6,15)(7,9)(8,11), (1,5)(2,6)(3,7)(4,8)(9,12)(10,11)(13,16)(14,15), (1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(13,15)(14,16), (5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,4)(2,3)(5,8)(6,7)(9,16)(10,15)(11,14)(12,13) );

G=PermutationGroup([[(1,4),(2,3),(5,8),(6,7),(9,15),(10,16),(11,13),(12,14)], [(1,16),(2,14),(3,12),(4,10),(5,13),(6,15),(7,9),(8,11)], [(1,5),(2,6),(3,7),(4,8),(9,12),(10,11),(13,16),(14,15)], [(1,2),(3,4),(5,6),(7,8),(9,11),(10,12),(13,15),(14,16)], [(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,4),(2,3),(5,8),(6,7),(9,16),(10,15),(11,14),(12,13)]])

G:=TransitiveGroup(16,271);

32 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	2J	···	2S	4A	···	4F	4G	···	4L
order	1	2	2	2	2	···	2	2	···	2	4	···	4	4	···	4
size	1	1	1	1	2	···	2	4	···	4	4	···	4	8	···	8

32 irreducible representations

dim	1	1	1	1	1	2	2	2	4
type	+	+	+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	D₄	D₄	D₄	C₂≀C₂²
kernel	C₂×C₂≀C₂²	C₂×C₂³⋊C₄	C₂≀C₂²	C₂×C₂²≀C₂	C₂×2₊¹⁺⁴	C₂²×C₄	C₂×D₄	C₂⁴	C₂
# reps	1	3	8	3	1	3	6	3	4

Matrix representation of C₂×C₂≀C₂² ►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

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

C₂×C₂≀C₂² in GAP, Magma, Sage, TeX

C_2\times C_2\wr C_2^2

% in TeX

G:=Group("C2xC2wrC2^2");

// GroupNames label

G:=SmallGroup(128,1755);

// by ID

G=gap.SmallGroup(128,1755);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,718,2028]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^4=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,e*b*e^-1=f*b*f=b*c*d,e*c*e^-1=c*d=d*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;

// generators/relations

G = C2×C2≀C22 order 128 = 27

Direct product of C2 and C2≀C22

G = C₂×C₂≀C₂² order 128 = 2⁷

Direct product of C₂ and C₂≀C₂²