(C2^2xC8):C4 - GroupNames

G = (C₂²×C₈)⋊C₄ order 128 = 2⁷

4^th semidirect product of C₂²×C₈ and C₄ acting faithfully

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

Aliases: (C₂²×C₈)⋊₄C₄, (C₂×D₄).₄Q₈, (C₂×D₄).₄₂D₄, C₂³⋊C₄.₁C₄, (C₂×C₄).₃C₄², C₄.₁₀D₄⋊₆C₄, C₂³.₃(C₄⋊C₄), (C₂×M₄(2))⋊₄C₄, (C₂²×C₄).₃₉D₄, C₄.₄₃(C₂³⋊C₄), C₂³.₄(C₂²⋊C₄), C₂.₁₇(C₂³.₉D₄), C₂³.C₂³.₂C₂, C₂².₆(C₂.C₄²), M₄(2).₈C₂².₄C₂, (C₂×C₄).₄(C₄⋊C₄), (C₂×D₄).₄₇(C₂×C₄), (C₂×Q₈).₄₂(C₂×C₄), (C₂×C₄).₄(C₂²⋊C₄), (C₂²×C₄).₆₈(C₂×C₄), (C₂×C₄○D₄).₃C₂², (C₂²×C₈)⋊C₂.₁₂C₂, SmallGroup(128,127)

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

Derived series

C₁ — C₂×C₄ — (C₂²×C₈)⋊C₄

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂×C₄○D₄ — (C₂²×C₈)⋊C₂ — (C₂²×C₈)⋊C₄

Lower central

C₁ — C₂ — C₂² — C₂×C₄ — (C₂²×C₈)⋊C₄

Upper central

C₁ — C₄ — C₂×C₄ — C₂×C₄○D₄ — (C₂²×C₈)⋊C₄

Jennings

C₁ — C₂ — C₂² — C₂×C₄○D₄ — (C₂²×C₈)⋊C₄

Generators and relations for (C₂²×C₈)⋊C₄
G = < a,b,c,d | a²=b²=c⁸=d⁴=1, dad^-1=ab=ba, ac=ca, bc=cb, dbd^-1=bc⁴, dcd^-1=abc³ >

Subgroups: 192 in 81 conjugacy classes, 30 normal (26 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, M₄(2), C₂²×C₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂²⋊C₈, C₂³⋊C₄, C₂³⋊C₄, C₄.D₄, C₄.₁₀D₄, C₄²⋊C₂, C₂²×C₈, C₂×M₄(2), C₂×M₄(2), C₂×C₄○D₄, (C₂²×C₈)⋊C₂, C₂³.C₂³, M₄(2).₈C₂², (C₂²×C₈)⋊C₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, Q₈, C₄², C₂²⋊C₄, C₄⋊C₄, C₂.C₄², C₂³⋊C₄, C₂³.₉D₄, (C₂²×C₈)⋊C₄

Character table of (C₂²×C₈)⋊C₄

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J
size	1	1	2	4	4	4	1	1	2	4	4	4	8	8	8	8	4	4	4	4	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	-i	i	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	i	-i	-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	i	i	-i	-i	-1	-1	-1	-1	1	-i	-i	i	i	1	linear of order 4
ρ₈	1	1	1	-1	1	-1	1	1	1	-1	1	-1	i	i	-i	-i	1	1	1	1	-1	i	i	-i	-i	-1	linear of order 4
ρ₉	1	1	1	-1	-1	-1	-1	-1	-1	1	1	1	i	-i	-i	i	i	-i	-i	i	-i	-1	1	-1	1	i	linear of order 4
ρ₁₀	1	1	1	-1	-1	-1	-1	-1	-1	1	1	1	-i	i	i	-i	-i	i	i	-i	i	-1	1	-1	1	-i	linear of order 4
ρ₁₁	1	1	1	-1	-1	-1	-1	-1	-1	1	1	1	-i	i	i	-i	i	-i	-i	i	-i	1	-1	1	-1	i	linear of order 4
ρ₁₂	1	1	1	-1	-1	-1	-1	-1	-1	1	1	1	i	-i	-i	i	-i	i	i	-i	i	1	-1	1	-1	-i	linear of order 4
ρ₁₃	1	1	1	-1	1	-1	1	1	1	-1	1	-1	-i	-i	i	i	1	1	1	1	-1	-i	-i	i	i	-1	linear of order 4
ρ₁₄	1	1	1	-1	1	-1	1	1	1	-1	1	-1	-i	-i	i	i	-1	-1	-1	-1	1	i	i	-i	-i	1	linear of order 4
ρ₁₅	1	1	1	1	-1	1	-1	-1	-1	-1	1	-1	-1	1	-1	1	-i	i	i	-i	-i	i	-i	-i	i	i	linear of order 4
ρ₁₆	1	1	1	1	-1	1	-1	-1	-1	-1	1	-1	1	-1	1	-1	i	-i	-i	i	i	i	-i	-i	i	-i	linear of order 4
ρ₁₇	2	2	2	-2	2	2	-2	-2	-2	2	-2	-2	0	0	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	-2	-2	0	0	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	-2	2	0	0	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	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from Q₈, Schur index 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 C₂³⋊C₄
ρ₂₃	4	-4	0	0	0	0	-4i	4i	0	0	0	0	0	0	0	0	2ζ₈	2ζ₈³	2ζ₈⁷	2ζ₈⁵	0	0	0	0	0	0	complex faithful
ρ₂₄	4	-4	0	0	0	0	4i	-4i	0	0	0	0	0	0	0	0	2ζ₈³	2ζ₈	2ζ₈⁵	2ζ₈⁷	0	0	0	0	0	0	complex faithful
ρ₂₅	4	-4	0	0	0	0	-4i	4i	0	0	0	0	0	0	0	0	2ζ₈⁵	2ζ₈⁷	2ζ₈³	2ζ₈	0	0	0	0	0	0	complex faithful
ρ₂₆	4	-4	0	0	0	0	4i	-4i	0	0	0	0	0	0	0	0	2ζ₈⁷	2ζ₈⁵	2ζ₈	2ζ₈³	0	0	0	0	0	0	complex faithful

Smallest permutation representation of (C₂²×C₈)⋊C₄
►On 32 points

Generators in S₃₂

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

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

(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)

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

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

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

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

Matrix representation of (C₂²×C₈)⋊C₄ ►in GL₄(𝔽₁₇) generated by

0	0	13	8
0	0	0	4
4	9	0	0
0	13	0	0

1	15	0	0
0	16	0	0
0	0	1	15
0	0	0	16

0	8	9	8
0	8	0	0
8	9	0	8
0	0	0	8

1	0	0	0
1	16	0	0
0	0	16	2
0	0	16	1

G:=sub<GL(4,GF(17))| [0,0,4,0,0,0,9,13,13,0,0,0,8,4,0,0],[1,0,0,0,15,16,0,0,0,0,1,0,0,0,15,16],[0,0,8,0,8,8,9,0,9,0,0,0,8,0,8,8],[1,1,0,0,0,16,0,0,0,0,16,16,0,0,2,1] >;

(C₂²×C₈)⋊C₄ in GAP, Magma, Sage, TeX

(C_2^2\times C_8)\rtimes C_4

% in TeX

G:=Group("(C2^2xC8):C4");

// GroupNames label

G:=SmallGroup(128,127);

// by ID

G=gap.SmallGroup(128,127);

# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,758,352,1466,521,2804,172,4037]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^8=d^4=1,d*a*d^-1=a*b=b*a,a*c=c*a,b*c=c*b,d*b*d^-1=b*c^4,d*c*d^-1=a*b*c^3>;

// generators/relations

Export

Character table of (C₂²×C₈)⋊C₄ in TeX

G = (C22×C8)⋊C4 order 128 = 27

4th semidirect product of C22×C8 and C4 acting faithfully

G = (C₂²×C₈)⋊C₄ order 128 = 2⁷

4^th semidirect product of C₂²×C₈ and C₄ acting faithfully