C2^2.45C2^4 - GroupNames

G = C₂².₄₅C₂⁴ order 64 = 2⁶

31^st central stem extension by C₂² of C₂⁴

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

Aliases: C₄²⋊₁₀C₂², C₂⁴.₂₀C₂², C₂³.₁₈C₂³, C₂².₄₅C₂⁴, C₂.₁₅2₊¹⁺⁴, (C₄×D₄)⋊₁₉C₂, C₄⋊C₄⋊₁₇C₂², C₂²⋊Q₈⋊₁₅C₂, (C₂×Q₈)⋊₅C₂², C₄²⋊₂C₂⋊₄C₂, C₂²≀C₂.₂C₂, C₄.₄D₄⋊₁₂C₂, C₂²⋊C₄⋊₈C₂², (C₂×C₄).₃₀C₂³, (C₂²×C₄)⋊₄C₂², C₄²⋊C₂⋊₁₄C₂, (C₂×D₄).₃₄C₂², C₂².₉(C₄○D₄), C₂².D₄⋊₁₀C₂, C₂.₂₄(C₂×C₄○D₄), (C₂×C₂²⋊C₄)⋊₁₅C₂, SmallGroup(64,232)

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

Derived series

C₁ — C₂² — C₂².₄₅C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂×C₂²⋊C₄ — C₂².₄₅C₂⁴

Lower central

C₁ — C₂² — C₂².₄₅C₂⁴

Upper central

C₁ — C₂² — C₂².₄₅C₂⁴

Jennings

C₁ — C₂² — C₂².₄₅C₂⁴

Generators and relations for C₂².₄₅C₂⁴
G = < a,b,c,d,e,f | a²=b²=e²=f²=1, c²=a, d²=b, ab=ba, dcd^-1=ac=ca, fdf=ad=da, ae=ea, af=fa, ece=bc=cb, bd=db, be=eb, bf=fb, cf=fc, de=ed, ef=fe >

Subgroups: 197 in 124 conjugacy classes, 75 normal (17 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂⁴, C₂×C₂²⋊C₄, C₄²⋊C₂, C₄×D₄, C₂²≀C₂, C₂²⋊Q₈, C₂².D₄, C₂².D₄, C₄.₄D₄, C₄²⋊₂C₂, C₂².₄₅C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, C₂².₄₅C₂⁴

Character table of C₂².₄₅C₂⁴

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L	4M	4N	4O
size	1	1	1	1	2	2	2	2	4	4	2	2	2	2	2	2	2	2	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	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	-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	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	-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	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	-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	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	-1	1	-1	linear of order 2
ρ₉	1	1	1	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
ρ₁₀	1	1	1	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
ρ₁₁	1	1	1	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
ρ₁₂	1	1	1	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
ρ₁₃	1	1	1	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
ρ₁₄	1	1	1	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
ρ₁₅	1	1	1	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
ρ₁₆	1	1	1	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
ρ₁₇	2	-2	2	-2	0	2	0	-2	0	0	0	0	-2i	-2i	2i	2i	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₈	2	-2	2	-2	0	-2	0	2	0	0	0	0	2i	-2i	-2i	2i	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₉	2	-2	2	-2	0	-2	0	2	0	0	0	0	-2i	2i	2i	-2i	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₀	2	2	-2	-2	2	0	-2	0	0	0	2i	-2i	0	0	0	0	-2i	2i	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₁	2	2	-2	-2	-2	0	2	0	0	0	2i	-2i	0	0	0	0	2i	-2i	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₂	2	2	-2	-2	-2	0	2	0	0	0	-2i	2i	0	0	0	0	-2i	2i	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₃	2	2	-2	-2	2	0	-2	0	0	0	-2i	2i	0	0	0	0	2i	-2i	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₄	2	-2	2	-2	0	2	0	-2	0	0	0	0	2i	2i	-2i	-2i	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₅	4	-4	-4	4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from 2₊¹⁺⁴

Permutation representations of C₂².₄₅C₂⁴
►On 16 points - transitive group 16T81

Generators in S₁₆

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

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

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

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

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

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

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

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

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

G:=TransitiveGroup(16,81);

Matrix representation of C₂².₄₅C₂⁴ ►in GL₄(𝔽₅) generated by

4	0	0	0
0	4	0	0
0	0	1	0
0	0	0	1

1	0	0	0
0	1	0	0
0	0	4	0
0	0	0	4

3	0	0	0
2	2	0	0
0	0	0	1
0	0	1	0

4	3	0	0
0	1	0	0
0	0	2	0
0	0	0	2

1	0	0	0
0	1	0	0
0	0	1	0
0	0	0	4

1	0	0	0
4	4	0	0
0	0	4	0
0	0	0	4

G:=sub<GL(4,GF(5))| [4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[3,2,0,0,0,2,0,0,0,0,0,1,0,0,1,0],[4,0,0,0,3,1,0,0,0,0,2,0,0,0,0,2],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,4],[1,4,0,0,0,4,0,0,0,0,4,0,0,0,0,4] >;

C₂².₄₅C₂⁴ in GAP, Magma, Sage, TeX

C_2^2._{45}C_2^4

% in TeX

G:=Group("C2^2.45C2^4");

// GroupNames label

G:=SmallGroup(64,232);

// by ID

G=gap.SmallGroup(64,232);

# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,96,217,199,650,297]);

// Polycyclic

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

// generators/relations

Export

Character table of C₂².₄₅C₂⁴ in TeX

G = C22.45C24 order 64 = 26

31st central stem extension by C22 of C24

G = C₂².₄₅C₂⁴ order 64 = 2⁶

31^st central stem extension by C₂² of C₂⁴