C2^4.C8 - GroupNames

G = C₂⁴.C₈ order 128 = 2⁷

2^nd non-split extension by C₂⁴ of C₈ acting via C₈/C₂=C₄

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C₂⁴.₂C₈, (C₂×C₈).₁₅D₄, C₂³.C₈⋊₄C₂, (C₂²×C₄).₂C₈, C₂³.₁₅(C₂×C₈), (C₂³×C₄).₁₂C₄, C₄.₃₇(C₂³⋊C₄), C₂.₁₀(C₂³⋊C₈), (C₂×M₄(2)).₆C₄, (C₂×C₄).₂₈M₄(2), C₄.₁₃(C₄.D₄), C₂².₁₅(C₂²⋊C₈), C₂⁴.₄C₄.₁₀C₂, (C₂×M₄(2)).₁₄₃C₂², (C₂²×C₄).₅₈(C₂×C₄), (C₂×C₄).₃₄₂(C₂²⋊C₄), SmallGroup(128,52)

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

Derived series

C₁ — C₂³ — C₂⁴.C₈

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂×C₈ — C₂×M₄(2) — C₂⁴.₄C₄ — C₂⁴.C₈

Lower central

C₁ — C₂ — C₂² — C₂³ — C₂⁴.C₈

Upper central

C₁ — C₄ — C₂×C₄ — C₂×M₄(2) — C₂⁴.C₈

Jennings

C₁ — C₂ — C₂ — C₂ — C₂ — C₄ — C₂×C₄ — C₂×M₄(2) — C₂⁴.C₈

Generators and relations for C₂⁴.C₈
G = < a,b,c,d,e | a²=b²=c²=d²=1, e⁸=d, ab=ba, ac=ca, ad=da, eae^-1=abcd, bc=cb, bd=db, ebe^-1=bcd, ece^-1=cd=dc, de=ed >

Subgroups: 160 in 64 conjugacy classes, 20 normal (16 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂³, C₂³, C₁₆, C₂×C₈, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂⁴, C₂²⋊C₈, M₅(2), C₂×M₄(2), C₂³×C₄, C₂³.C₈, C₂⁴.₄C₄, C₂⁴.C₈
Quotients: C₁, C₂, C₄, C₂², C₈, C₂×C₄, D₄, C₂²⋊C₄, C₂×C₈, M₄(2), C₂²⋊C₈, C₂³⋊C₄, C₄.D₄, C₂³⋊C₈, C₂⁴.C₈

Character table of C₂⁴.C₈

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	8A	8B	8C	8D	8E	8F	16A	16B	16C	16D	16E	16F	16G	16H
size	1	1	2	4	4	4	1	1	2	4	4	4	4	4	4	4	8	8	8	8	8	8	8	8	8	8
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	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	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	-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	-1	linear of order 2
ρ₅	1	1	1	-1	-1	1	1	1	1	-1	-1	1	-1	-1	-1	-1	1	1	-i	-i	i	-i	i	-i	i	i	linear of order 4
ρ₆	1	1	1	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	i	i	-i	-i	-i	-i	i	i	linear of order 4
ρ₇	1	1	1	-1	-1	1	1	1	1	-1	-1	1	-1	-1	-1	-1	1	1	i	i	-i	i	-i	i	-i	-i	linear of order 4
ρ₈	1	1	1	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	-i	-i	i	i	i	i	-i	-i	linear of order 4
ρ₉	1	1	1	-1	-1	1	-1	-1	-1	1	1	-1	i	i	-i	-i	i	-i	ζ₈⁵	ζ₈	ζ₈⁷	ζ₈³	ζ₈³	ζ₈⁷	ζ₈⁵	ζ₈	linear of order 8
ρ₁₀	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	-i	-i	i	i	i	-i	ζ₈⁷	ζ₈³	ζ₈⁵	ζ₈⁵	ζ₈	ζ₈	ζ₈³	ζ₈⁷	linear of order 8
ρ₁₁	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	-i	-i	i	i	i	-i	ζ₈³	ζ₈⁷	ζ₈	ζ₈	ζ₈⁵	ζ₈⁵	ζ₈⁷	ζ₈³	linear of order 8
ρ₁₂	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	i	i	-i	-i	-i	i	ζ₈	ζ₈⁵	ζ₈³	ζ₈³	ζ₈⁷	ζ₈⁷	ζ₈⁵	ζ₈	linear of order 8
ρ₁₃	1	1	1	-1	-1	1	-1	-1	-1	1	1	-1	i	i	-i	-i	i	-i	ζ₈	ζ₈⁵	ζ₈³	ζ₈⁷	ζ₈⁷	ζ₈³	ζ₈	ζ₈⁵	linear of order 8
ρ₁₄	1	1	1	-1	-1	1	-1	-1	-1	1	1	-1	-i	-i	i	i	-i	i	ζ₈³	ζ₈⁷	ζ₈	ζ₈⁵	ζ₈⁵	ζ₈	ζ₈³	ζ₈⁷	linear of order 8
ρ₁₅	1	1	1	-1	-1	1	-1	-1	-1	1	1	-1	-i	-i	i	i	-i	i	ζ₈⁷	ζ₈³	ζ₈⁵	ζ₈	ζ₈	ζ₈⁵	ζ₈⁷	ζ₈³	linear of order 8
ρ₁₆	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	i	i	-i	-i	-i	i	ζ₈⁵	ζ₈	ζ₈⁷	ζ₈⁷	ζ₈³	ζ₈³	ζ₈	ζ₈⁵	linear of order 8
ρ₁₇	2	2	2	0	0	-2	2	2	2	0	0	-2	-2	2	-2	2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₈	2	2	2	0	0	-2	2	2	2	0	0	-2	2	-2	2	-2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₉	2	2	2	0	0	-2	-2	-2	-2	0	0	2	2i	-2i	-2i	2i	0	0	0	0	0	0	0	0	0	0	complex lifted from M₄(2)
ρ₂₀	2	2	2	0	0	-2	-2	-2	-2	0	0	2	-2i	2i	2i	-2i	0	0	0	0	0	0	0	0	0	0	complex lifted from M₄(2)
ρ₂₁	4	4	-4	0	0	0	4	4	-4	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	0	0	0	-4	-4	4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₄.D₄
ρ₂₃	4	-4	0	-2	2	0	4i	-4i	0	-2i	2i	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	complex faithful
ρ₂₄	4	-4	0	-2	2	0	-4i	4i	0	2i	-2i	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	complex faithful
ρ₂₅	4	-4	0	2	-2	0	-4i	4i	0	-2i	2i	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	complex faithful
ρ₂₆	4	-4	0	2	-2	0	4i	-4i	0	2i	-2i	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	complex faithful

Permutation representations of C₂⁴.C₈
►On 16 points - transitive group 16T306

Generators in S₁₆

(1 9)(3 11)(4 12)(5 13)(7 15)(8 16)

(2 10)(3 11)(6 14)(7 15)

(2 10)(4 12)(6 14)(8 16)

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

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

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

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

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

G:=TransitiveGroup(16,306);

Matrix representation of C₂⁴.C₈ ►in GL₄(𝔽₅) generated by

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

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

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

4	0	0	0
0	4	0	0
0	0	4	0
0	0	0	4

0	0	0	3
1	0	0	0
0	1	0	0
0	0	1	0

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

C₂⁴.C₈ in GAP, Magma, Sage, TeX

C_2^4.C_8

% in TeX

G:=Group("C2^4.C8");

// GroupNames label

G:=SmallGroup(128,52);

// by ID

G=gap.SmallGroup(128,52);

# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,422,346,521,136,2804,124]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^2=1,e^8=d,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b*c*d,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 C₂⁴.C₈ in TeX

G = C24.C8 order 128 = 27

2nd non-split extension by C24 of C8 acting via C8/C2=C4

G = C₂⁴.C₈ order 128 = 2⁷

2^nd non-split extension by C₂⁴ of C₈ acting via C₈/C₂=C₄