C4^2:3C8 - GroupNames

G = C₄²⋊₃C₈ order 128 = 2⁷

3^rd semidirect product of C₄² and C₈ acting via C₈/C₂=C₄

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

Aliases: C₄²⋊₃C₈, (C₂×C₄²).₉C₄, (C₂²×C₄).₆D₄, C₄²⋊₉C₄.₂C₂, (C₂×C₄).₃₀M₄(2), C₂.₂(C₄²⋊C₄), C₂².₁₈(C₂²⋊C₈), C₂².₄₀(C₂³⋊C₄), C₂.₂(C₄².₃C₄), C₂³.₁₅₂(C₂²⋊C₄), C₂².₇(C₄.₁₀D₄), C₂².M₄(2).₃C₂, C₂.₆(C₂².M₄(2)), (C₂×C₄⋊C₄).₇C₄, (C₂×C₄).₃₅(C₂×C₈), (C₂×C₄⋊C₄).₄C₂², (C₂²×C₄).₆₁(C₂×C₄), SmallGroup(128,57)

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

Derived series

C₁ — C₂×C₄ — C₄²⋊₃C₈

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄⋊C₄ — C₄²⋊₉C₄ — C₄²⋊₃C₈

Lower central

C₁ — C₂ — C₂² — C₂×C₄ — C₄²⋊₃C₈

Upper central

C₁ — C₂² — C₂³ — C₂×C₄⋊C₄ — C₄²⋊₃C₈

Jennings

C₁ — C₂² — C₂³ — C₂×C₄⋊C₄ — C₄²⋊₃C₈

Generators and relations for C₄²⋊₃C₈
G = < a,b,c | a⁴=b⁴=c⁸=1, ab=ba, cac^-1=a^-1b^-1, cbc^-1=a²b >

Subgroups: 160 in 67 conjugacy classes, 22 normal (16 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂³, C₄², C₄², C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂²⋊C₈, C₂×C₄², C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂².M₄(2), C₄²⋊₉C₄, C₄²⋊₃C₈
Quotients: C₁, C₂, C₄, C₂², C₈, C₂×C₄, D₄, C₂²⋊C₄, C₂×C₈, M₄(2), C₂²⋊C₈, C₂³⋊C₄, C₄.₁₀D₄, C₂².M₄(2), C₄²⋊C₄, C₄².₃C₄, C₄²⋊₃C₈

Character table of C₄²⋊₃C₈

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L	8A	8B	8C	8D	8E	8F	8G	8H
size	1	1	1	1	2	2	4	4	4	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	-i	-1	-i	i	1	1	i	i	-i	ζ₈⁷	ζ₈⁵	ζ₈³	ζ₈³	ζ₈⁵	ζ₈⁷	ζ₈	ζ₈	linear of order 8
ρ₁₀	1	-1	1	-1	1	-1	1	1	1	i	-1	i	-i	-1	-1	-i	i	-i	ζ₈	ζ₈³	ζ₈	ζ₈⁵	ζ₈⁷	ζ₈⁵	ζ₈³	ζ₈⁷	linear of order 8
ρ₁₁	1	-1	1	-1	1	-1	-1	1	-1	i	-1	i	-i	1	1	-i	-i	i	ζ₈⁵	ζ₈⁷	ζ₈	ζ₈	ζ₈⁷	ζ₈⁵	ζ₈³	ζ₈³	linear of order 8
ρ₁₂	1	-1	1	-1	1	-1	-1	1	-1	-i	-1	-i	i	1	1	i	i	-i	ζ₈³	ζ₈	ζ₈⁷	ζ₈⁷	ζ₈	ζ₈³	ζ₈⁵	ζ₈⁵	linear of order 8
ρ₁₃	1	-1	1	-1	1	-1	1	1	1	-i	-1	-i	i	-1	-1	i	-i	i	ζ₈⁷	ζ₈⁵	ζ₈⁷	ζ₈³	ζ₈	ζ₈³	ζ₈⁵	ζ₈	linear of order 8
ρ₁₄	1	-1	1	-1	1	-1	1	1	1	i	-1	i	-i	-1	-1	-i	i	-i	ζ₈⁵	ζ₈⁷	ζ₈⁵	ζ₈	ζ₈³	ζ₈	ζ₈⁷	ζ₈³	linear of order 8
ρ₁₅	1	-1	1	-1	1	-1	-1	1	-1	i	-1	i	-i	1	1	-i	-i	i	ζ₈	ζ₈³	ζ₈⁵	ζ₈⁵	ζ₈³	ζ₈	ζ₈⁷	ζ₈⁷	linear of order 8
ρ₁₆	1	-1	1	-1	1	-1	1	1	1	-i	-1	-i	i	-1	-1	i	-i	i	ζ₈³	ζ₈	ζ₈³	ζ₈⁷	ζ₈⁵	ζ₈⁷	ζ₈	ζ₈⁵	linear of order 8
ρ₁₇	2	2	2	2	2	2	0	-2	0	2	-2	-2	-2	0	0	2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₈	2	2	2	2	2	2	0	-2	0	-2	-2	2	2	0	0	-2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₉	2	-2	2	-2	2	-2	0	-2	0	2i	2	-2i	2i	0	0	-2i	0	0	0	0	0	0	0	0	0	0	complex lifted from M₄(2)
ρ₂₀	2	-2	2	-2	2	-2	0	-2	0	-2i	2	2i	-2i	0	0	2i	0	0	0	0	0	0	0	0	0	0	complex lifted from M₄(2)
ρ₂₁	4	4	4	4	-4	-4	0	0	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	2	0	-2	0	0	0	0	2	-2	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₄²⋊C₄
ρ₂₃	4	-4	-4	4	0	0	-2	0	2	0	0	0	0	-2	2	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₄²⋊C₄
ρ₂₄	4	4	-4	-4	0	0	2	0	-2	0	0	0	0	-2	2	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₄².₃C₄, Schur index 2
ρ₂₅	4	-4	4	-4	-4	4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₄.₁₀D₄, Schur index 2
ρ₂₆	4	4	-4	-4	0	0	-2	0	2	0	0	0	0	2	-2	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₄².₃C₄, Schur index 2

Smallest permutation representation of C₄²⋊₃C₈
►On 32 points

Generators in S₃₂

(2 23 32 16)(4 10 26 17)(6 19 28 12)(8 14 30 21)

(1 15 31 22)(2 16 32 23)(3 24 25 9)(4 17 26 10)(5 11 27 18)(6 12 28 19)(7 20 29 13)(8 21 30 14)

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)

G:=sub<Sym(32)| (2,23,32,16)(4,10,26,17)(6,19,28,12)(8,14,30,21), (1,15,31,22)(2,16,32,23)(3,24,25,9)(4,17,26,10)(5,11,27,18)(6,12,28,19)(7,20,29,13)(8,21,30,14), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)>;

G:=Group( (2,23,32,16)(4,10,26,17)(6,19,28,12)(8,14,30,21), (1,15,31,22)(2,16,32,23)(3,24,25,9)(4,17,26,10)(5,11,27,18)(6,12,28,19)(7,20,29,13)(8,21,30,14), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32) );

G=PermutationGroup([[(2,23,32,16),(4,10,26,17),(6,19,28,12),(8,14,30,21)], [(1,15,31,22),(2,16,32,23),(3,24,25,9),(4,17,26,10),(5,11,27,18),(6,12,28,19),(7,20,29,13),(8,21,30,14)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32)]])

Matrix representation of C₄²⋊₃C₈ ►in GL₆(𝔽₁₇)

1	0	0	0	0	0
16	16	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	5	14	10	14
0	0	7	6	11	7

16	0	0	0	0	0
0	16	0	0	0	0
0	0	16	10	0	0
0	0	10	1	0	0
0	0	12	3	7	3
0	0	0	3	6	10

9	1	0	0	0	0
0	8	0	0	0	0
0	0	0	0	1	0
0	0	9	15	13	15
0	0	0	1	0	0
0	0	0	0	0	2

G:=sub<GL(6,GF(17))| [1,16,0,0,0,0,0,16,0,0,0,0,0,0,1,0,5,7,0,0,0,1,14,6,0,0,0,0,10,11,0,0,0,0,14,7],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,10,12,0,0,0,10,1,3,3,0,0,0,0,7,6,0,0,0,0,3,10],[9,0,0,0,0,0,1,8,0,0,0,0,0,0,0,9,0,0,0,0,0,15,1,0,0,0,1,13,0,0,0,0,0,15,0,2] >;

C₄²⋊₃C₈ in GAP, Magma, Sage, TeX

C_4^2\rtimes_3C_8

% in TeX

G:=Group("C4^2:3C8");

// GroupNames label

G:=SmallGroup(128,57);

// by ID

G=gap.SmallGroup(128,57);

# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,568,422,352,1242,521,136,2804]);

// Polycyclic

G:=Group<a,b,c|a^4=b^4=c^8=1,a*b=b*a,c*a*c^-1=a^-1*b^-1,c*b*c^-1=a^2*b>;

// generators/relations

Export

Character table of C₄²⋊₃C₈ in TeX

G = C42⋊3C8 order 128 = 27

3rd semidirect product of C42 and C8 acting via C8/C2=C4

G = C₄²⋊₃C₈ order 128 = 2⁷

3^rd semidirect product of C₄² and C₈ acting via C₈/C₂=C₄