C2xC2^3:C4 - GroupNames

G = C₂×C₂³⋊C₄ order 64 = 2⁶

Direct product of C₂ and C₂³⋊C₄

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

Aliases: C₂×C₂³⋊C₄, C₂⁴⋊₂C₄, C₂³.₁₂D₄, C₂³.₁C₂³, C₂⁴.₁₀C₂², (C₂×D₄)⋊₅C₄, C₂³⋊₁(C₂×C₄), (C₂²×C₄)⋊₃C₄, C₂².₈(C₂×D₄), (C₂²×D₄).₄C₂, C₂²⋊C₄⋊₁₁C₂², (C₂×D₄).₄₂C₂², C₂².₆(C₂²×C₄), C₂².₂₉(C₂²⋊C₄), (C₂×C₄)⋊₁(C₂×C₄), (C₂×C₂²⋊C₄)⋊₄C₂, C₂.₁₂(C₂×C₂²⋊C₄), SmallGroup(64,90)

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

Derived series

C₁ — C₂² — C₂×C₂³⋊C₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂²×D₄ — C₂×C₂³⋊C₄

Lower central

C₁ — C₂ — C₂² — C₂×C₂³⋊C₄

Upper central

C₁ — C₂² — C₂⁴ — C₂×C₂³⋊C₄

Jennings

C₁ — C₂ — C₂³ — C₂×C₂³⋊C₄

Generators and relations for C₂×C₂³⋊C₄
G = < a,b,c,d,e | a²=b²=c²=d²=e⁴=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe^-1=bcd, ece^-1=cd=dc, de=ed >

Subgroups: 217 in 105 conjugacy classes, 41 normal (13 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₂²⋊C₄, C₂²⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂³⋊C₄, C₂×C₂²⋊C₄, C₂²×D₄, C₂×C₂³⋊C₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂³⋊C₄, C₂×C₂²⋊C₄, C₂×C₂³⋊C₄

Character table of C₂×C₂³⋊C₄

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	2J	2K	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J
size	1	1	1	1	2	2	2	2	2	2	4	4	4	4	4	4	4	4	4	4	4	4
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	1	1	-1	-1	1	1	linear of order 2
ρ₃	1	-1	1	-1	-1	1	1	1	-1	-1	1	-1	-1	1	-1	1	1	-1	-1	1	-1	1	linear of order 2
ρ₄	1	-1	1	-1	-1	1	1	1	-1	-1	-1	1	1	-1	1	-1	1	-1	1	-1	-1	1	linear of order 2
ρ₅	1	1	1	1	1	1	1	1	1	1	1	1	-1	-1	1	1	-1	-1	-1	-1	-1	-1	linear of order 2
ρ₆	1	1	1	1	1	1	1	1	1	1	-1	-1	1	1	-1	-1	-1	-1	1	1	-1	-1	linear of order 2
ρ₇	1	-1	1	-1	-1	1	1	1	-1	-1	1	-1	1	-1	-1	1	-1	1	1	-1	1	-1	linear of order 2
ρ₈	1	-1	1	-1	-1	1	1	1	-1	-1	-1	1	-1	1	1	-1	-1	1	-1	1	1	-1	linear of order 2
ρ₉	1	-1	1	-1	1	-1	1	-1	1	-1	1	-1	i	-i	1	-1	i	-i	-i	i	i	-i	linear of order 4
ρ₁₀	1	-1	1	-1	1	-1	1	-1	1	-1	1	-1	-i	i	1	-1	-i	i	i	-i	-i	i	linear of order 4
ρ₁₁	1	1	1	1	-1	-1	1	-1	-1	1	1	1	-i	-i	-1	-1	i	i	i	i	-i	-i	linear of order 4
ρ₁₂	1	1	1	1	-1	-1	1	-1	-1	1	1	1	i	i	-1	-1	-i	-i	-i	-i	i	i	linear of order 4
ρ₁₃	1	-1	1	-1	1	-1	1	-1	1	-1	-1	1	i	-i	-1	1	-i	i	-i	i	-i	i	linear of order 4
ρ₁₄	1	-1	1	-1	1	-1	1	-1	1	-1	-1	1	-i	i	-1	1	i	-i	i	-i	i	-i	linear of order 4
ρ₁₅	1	1	1	1	-1	-1	1	-1	-1	1	-1	-1	-i	-i	1	1	-i	-i	i	i	i	i	linear of order 4
ρ₁₆	1	1	1	1	-1	-1	1	-1	-1	1	-1	-1	i	i	1	1	i	i	-i	-i	-i	-i	linear of order 4
ρ₁₇	2	2	2	2	2	-2	-2	2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₈	2	-2	2	-2	2	2	-2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₉	2	-2	2	-2	-2	-2	-2	2	2	2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₂₀	2	2	2	2	-2	2	-2	-2	2	-2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₂₁	4	4	-4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₂³⋊C₄
ρ₂₂	4	-4	-4	4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₂³⋊C₄

Permutation representations of C₂×C₂³⋊C₄
►On 16 points - transitive group 16T76

Generators in S₁₆

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

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

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

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

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

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

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

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

G:=TransitiveGroup(16,76);

►On 16 points - transitive group 16T78

Generators in S₁₆

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

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

(2 14)(4 16)(5 11)(7 9)

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

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

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

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

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

G:=TransitiveGroup(16,78);

►On 16 points - transitive group 16T92

Generators in S₁₆

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

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

(2 4)(6 7)(10 12)(13 15)

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

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

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

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

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

G:=TransitiveGroup(16,92);

►On 16 points - transitive group 16T93

Generators in S₁₆

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

(1 5)(3 9)(4 14)(7 13)(8 10)(11 15)

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

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

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

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

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

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

G:=TransitiveGroup(16,93);

►On 16 points - transitive group 16T94

Generators in S₁₆

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

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

(2 15)(4 13)(5 10)(7 12)

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

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

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

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

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

G:=TransitiveGroup(16,94);

►On 16 points - transitive group 16T102

Generators in S₁₆

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

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

(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)

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

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

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

G:=TransitiveGroup(16,102);

C₂×C₂³⋊C₄ is a maximal subgroup of
C₂⁴.₅D₄ C₂⁴.₆D₄ C₂⁵⋊C₄ C₂⁴.₁₆₅C₂³ 2₊¹⁺⁴⋊₂C₄ C₂⁴.₁₆₇C₂³ C₂⁴.C₂³ C₂⁴.₂₂D₄ C₂⁵.C₂² C₂⁴.₂₆D₄ C₂⁴.₇₈D₄ C₂⁴.₁₇₄C₂³ C₂⁴.₂₈D₄ C₂⁴.₁₇₅C₂³ C₂⁴⋊D₄ C₂⁴.₃₁D₄ C₂⁴⋊Q₈ C₂⁴.₃₃D₄ C₂⁴.₃₆D₄ C₂⁴.₃₉D₄ C₂³.C₂⁴ C₂⁴⋊C₂³ (C₂×D₄)⋊₇F₅
C₂×C₂³⋊C₄ is a maximal quotient of
C₂³.₈M₄(2) C₂⁵.₃C₄ (C₂×C₄)⋊M₄(2) C₂³⋊M₄(2) C₂³⋊C₈⋊C₂ C₂⁴.(C₂×C₄) C₂⁴.₄₅(C₂×C₄) C₂⁴.₅₃D₄ C₂⁴.₁₅₀D₄ C₂⁴.₅₄D₄ C₂⁴.₅₅D₄ C₂⁴.₅₆D₄ C₂⁴.₅₇D₄ C₂⁴.₅₈D₄ C₂⁴.₅₉D₄ C₂⁴.₆₀D₄ C₂⁴.₆₁D₄ C₂⁵⋊C₄ C₂⁴.₁₆₇C₂³ C₂⁴.₆₈D₄ C₂⁴.₇₈D₄ C₂⁴.₁₇₅C₂³ C₂⁴.₁₇₆C₂³ C₄○C₂≀C₄ C₂⁴.₃₆D₄ C₂≀C₄⋊C₂ C₂³.(C₂×D₄) C₄⋊Q₈⋊₂₉C₄ C₂⁴.₃₉D₄ C₄.₄D₄⋊C₄ C₄⋊Q₈⋊C₄ (C₂×D₄).₁₃₅D₄ C₄⋊Q₈.C₄ C₄⋊₁D₄.C₄ (C₂×D₄).₁₃₇D₄ (C₂×D₄)⋊₇F₅

Matrix representation of C₂×C₂³⋊C₄ ►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	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

0	-1	0	0	0	0
1	0	0	0	0	0
0	0	0	0	-1	0
0	0	0	0	0	-1
0	0	0	-1	0	0
0	0	-1	0	0	0

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

C₂×C₂³⋊C₄ in GAP, Magma, Sage, TeX

C_2\times C_2^3\rtimes C_4

% in TeX

G:=Group("C2xC2^3:C4");

// GroupNames label

G:=SmallGroup(64,90);

// by ID

G=gap.SmallGroup(64,90);

# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,963,730]);

// Polycyclic

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

// generators/relations

Export

Character table of C₂×C₂³⋊C₄ in TeX

G = C2×C23⋊C4 order 64 = 26

Direct product of C2 and C23⋊C4

G = C₂×C₂³⋊C₄ order 64 = 2⁶

Direct product of C₂ and C₂³⋊C₄