C2^2.32C2^4 - GroupNames

G = C₂².₃₂C₂⁴ order 64 = 2⁶

18^th central stem extension by C₂² of C₂⁴

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

Aliases: C₄²⋊₇C₂², C₂⁴.₁₇C₂², C₂³.₁₃C₂³, C₂².₃₂C₂⁴, C₂.₆2₊¹⁺⁴, (C₄×D₄)⋊₁₀C₂, C₄⋊D₄⋊₈C₂, C₂²⋊Q₈⋊₇C₂, C₂²≀C₂⋊₄C₂, C₄⋊C₄⋊₁₅C₂², C₄.₄D₄⋊₈C₂, (C₂×D₄)⋊₄C₂², (C₂×Q₈)⋊₃C₂², C₄²⋊₂C₂⋊₁C₂, C₂²⋊C₄⋊₅C₂², (C₂×C₄).₁₉C₂³, (C₂²×C₄)⋊₉C₂², C₂².₄(C₄○D₄), C₂².D₄⋊₄C₂, C₂.₁₅(C₂×C₄○D₄), (C₂×C₂²⋊C₄)⋊₁₃C₂, SmallGroup(64,219)

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

Derived series

C₁ — C₂² — C₂².₃₂C₂⁴

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂×C₂²⋊C₄ — C₂².₃₂C₂⁴

Lower central

C₁ — C₂² — C₂².₃₂C₂⁴

Upper central

C₁ — C₂² — C₂².₃₂C₂⁴

Jennings

C₁ — C₂² — C₂².₃₂C₂⁴

Generators and relations for C₂².₃₂C₂⁴
G = < a,b,c,d,e,f | a²=b²=c²=e²=f²=1, d²=b, ab=ba, dcd^-1=fcf=ac=ca, ede=ad=da, ae=ea, af=fa, ece=bc=cb, bd=db, be=eb, bf=fb, df=fd, ef=fe >

Subgroups: 217 in 125 conjugacy classes, 73 normal (19 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂⁴, C₂×C₂²⋊C₄, C₄×D₄, C₂²≀C₂, C₄⋊D₄, C₄⋊D₄, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄²⋊₂C₂, C₂².₃₂C₂⁴
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, C₂².₃₂C₂⁴

Character table of C₂².₃₂C₂⁴

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L
size	1	1	1	1	2	2	4	4	4	4	2	2	2	2	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	-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	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	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	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	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	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	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	linear of order 2
ρ₁₇	2	-2	2	-2	-2	2	0	0	0	0	2i	-2i	2i	-2i	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₈	2	-2	2	-2	2	-2	0	0	0	0	2i	-2i	-2i	2i	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₉	2	-2	2	-2	-2	2	0	0	0	0	-2i	2i	-2i	2i	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₀	2	-2	2	-2	2	-2	0	0	0	0	-2i	2i	2i	-2i	0	0	0	0	0	0	0	0	complex lifted from C₄○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 2₊¹⁺⁴
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 2₊¹⁺⁴

Permutation representations of C₂².₃₂C₂⁴
►On 16 points - transitive group 16T82

Generators in S₁₆

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

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

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

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

(1 3)(2 12)(4 10)(6 15)(8 13)(9 11)

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

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

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

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

G:=TransitiveGroup(16,82);

Matrix representation of C₂².₃₂C₂⁴ ►in GL₆(𝔽₅)

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

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

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

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

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	4	0
0	0	0	0	0	4

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4] >;

C₂².₃₂C₂⁴ in GAP, Magma, Sage, TeX

C_2^2._{32}C_2^4

% in TeX

G:=Group("C2^2.32C2^4");

// GroupNames label

G:=SmallGroup(64,219);

// by ID

G=gap.SmallGroup(64,219);

# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,217,199,650,188,579]);

// Polycyclic

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

// generators/relations

Export

Character table of C₂².₃₂C₂⁴ in TeX

G = C22.32C24 order 64 = 26

18th central stem extension by C22 of C24

G = C₂².₃₂C₂⁴ order 64 = 2⁶

18^th central stem extension by C₂² of C₂⁴