C2xC4^2.C4 - GroupNames

G = C₂xC₄².C₄ order 128 = 2⁷

Direct product of C₂ and C₄².C₄

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

Aliases: C₂xC₄².C₄, (C₂xC₄²).₂₃C₄, C₄².₂₅(C₂xC₄), (C₂xQ₈).₁₁₇D₄, (C₂²xC₄).₉₇D₄, C₄.₄D₄.₁₃C₄, (C₂²xD₄).₁₅C₄, (C₂xQ₈).₁₀C₂³, C₄.₁₀D₄:₁₇C₂², C₂².₅₂(C₂³:C₄), (C₂²xQ₈).₈₄C₂², C₂³.₂₀₃(C₂²:C₄), C₄.₄D₄.₁₂₂C₂², (C₂xC₄).₇(C₂xD₄), (C₂xD₄).₃₉(C₂xC₄), C₂.₄₂(C₂xC₂³:C₄), (C₂xC₄).₉₉(C₂²xC₄), (C₂²xC₄).₈₂(C₂xC₄), (C₂xQ₈).₁₀₈(C₂xC₄), (C₂xC₄.₁₀D₄):₂₆C₂, (C₂xC₄).₂₉(C₂²:C₄), (C₂xC₄.₄D₄).₁₄C₂, C₂².₆₆(C₂xC₂²:C₄), SmallGroup(128,862)

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

Derived series

C₁ — C₂xC₄ — C₂xC₄².C₄

Chief series

C₁ — C₂ — C₂² — C₂xC₄ — C₂xQ₈ — C₂²xQ₈ — C₂xC₄.₄D₄ — C₂xC₄².C₄

Lower central

C₁ — C₂ — C₂² — C₂xC₄ — C₂xC₄².C₄

Upper central

C₁ — C₂² — C₂³ — C₂²xQ₈ — C₂xC₄².C₄

Jennings

C₁ — C₂ — C₂² — C₂xQ₈ — C₂xC₄².C₄

Generators and relations for C₂xC₄².C₄
G = < a,b,c,d | a²=b⁴=c⁴=1, d⁴=c², ab=ba, ac=ca, ad=da, bc=cb, dbd^-1=b^-1c^-1, dcd^-1=b²c^-1 >

Subgroups: 324 in 130 conjugacy classes, 44 normal (16 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₈, C₂xC₄, C₂xC₄, C₂xC₄, D₄, Q₈, C₂³, C₂³, C₄², C₄², C₂²:C₄, C₂xC₈, M₄(2), C₂²xC₄, C₂²xC₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂xQ₈, C₂xQ₈, C₂xQ₈, C₂⁴, C₄.₁₀D₄, C₄.₁₀D₄, C₂xC₄², C₂xC₂²:C₄, C₄.₄D₄, C₄.₄D₄, C₂xM₄(2), C₂²xD₄, C₂²xQ₈, C₄².C₄, C₂xC₄.₁₀D₄, C₂xC₄.₄D₄, C₂xC₄².C₄
Quotients: C₁, C₂, C₄, C₂², C₂xC₄, D₄, C₂³, C₂²:C₄, C₂²xC₄, C₂xD₄, C₂³:C₄, C₂xC₂²:C₄, C₄².C₄, C₂xC₂³:C₄, C₂xC₄².C₄

Character table of C₂xC₄².C₄

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	8A	8B	8C	8D	8E	8F	8G	8H
size	1	1	1	1	2	2	8	8	4	4	4	4	4	4	4	4	4	4	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	-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	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	-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	-1	linear of order 2
ρ₉	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
ρ₁₀	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
ρ₁₁	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
ρ₁₂	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
ρ₁₃	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
ρ₁₄	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
ρ₁₅	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
ρ₁₆	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
ρ₁₇	2	2	2	2	2	2	0	0	0	2	0	0	-2	-2	2	-2	0	-2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₈	2	-2	2	-2	2	-2	0	0	0	-2	0	0	-2	-2	2	2	0	2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₉	2	-2	2	-2	2	-2	0	0	0	2	0	0	2	-2	-2	-2	0	2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₂₀	2	2	2	2	2	2	0	0	0	-2	0	0	2	-2	-2	2	0	-2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
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	-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
ρ₂₃	4	4	-4	-4	0	0	0	0	2i	0	2i	-2i	0	0	0	0	-2i	0	0	0	0	0	0	0	0	0	complex lifted from C₄².C₄
ρ₂₄	4	4	-4	-4	0	0	0	0	-2i	0	-2i	2i	0	0	0	0	2i	0	0	0	0	0	0	0	0	0	complex lifted from C₄².C₄
ρ₂₅	4	-4	-4	4	0	0	0	0	2i	0	-2i	-2i	0	0	0	0	2i	0	0	0	0	0	0	0	0	0	complex lifted from C₄².C₄
ρ₂₆	4	-4	-4	4	0	0	0	0	-2i	0	2i	2i	0	0	0	0	-2i	0	0	0	0	0	0	0	0	0	complex lifted from C₄².C₄

Smallest permutation representation of C₂xC₄².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 11 9 3)(2 18 14 26)(4 32 16 24)(5 15 13 7)(6 22 10 30)(8 28 12 20)(17 19 29 31)(21 23 25 27)

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

(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)| (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), (1,11,9,3)(2,18,14,26)(4,32,16,24)(5,15,13,7)(6,22,10,30)(8,28,12,20)(17,19,29,31)(21,23,25,27), (1,23,5,19)(2,32,6,28)(3,21,7,17)(4,30,8,26)(9,27,13,31)(10,20,14,24)(11,25,15,29)(12,18,16,22), (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( (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), (1,11,9,3)(2,18,14,26)(4,32,16,24)(5,15,13,7)(6,22,10,30)(8,28,12,20)(17,19,29,31)(21,23,25,27), (1,23,5,19)(2,32,6,28)(3,21,7,17)(4,30,8,26)(9,27,13,31)(10,20,14,24)(11,25,15,29)(12,18,16,22), (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([[(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)], [(1,11,9,3),(2,18,14,26),(4,32,16,24),(5,15,13,7),(6,22,10,30),(8,28,12,20),(17,19,29,31),(21,23,25,27)], [(1,23,5,19),(2,32,6,28),(3,21,7,17),(4,30,8,26),(9,27,13,31),(10,20,14,24),(11,25,15,29),(12,18,16,22)], [(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 C₂xC₄².C₄ ►in GL₆(F₁₇)

16	0	0	0	0	0
0	16	0	0	0	0
0	0	16	0	0	0
0	0	0	16	0	0
0	0	0	0	16	0
0	0	0	0	0	16

16	0	0	0	0	0
2	1	0	0	0	0
0	0	0	16	0	0
0	0	16	0	0	0
0	0	0	0	13	0
0	0	0	0	0	13

16	0	0	0	0	0
0	16	0	0	0	0
0	0	0	13	0	0
0	0	13	0	0	0
0	0	0	0	0	13
0	0	0	0	13	0

4	4	0	0	0	0
0	13	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	0	1	0	0
0	0	16	0	0	0

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,2,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,13,0],[4,0,0,0,0,0,4,13,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;

C₂xC₄².C₄ in GAP, Magma, Sage, TeX

C_2\times C_4^2.C_4

% in TeX

G:=Group("C2xC4^2.C4");

// GroupNames label

G:=SmallGroup(128,862);

// by ID

G=gap.SmallGroup(128,862);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,1123,1018,248,1971,375,172,4037]);

// Polycyclic

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

// generators/relations

Export

Character table of C₂xC₄².C₄ in TeX

G = C2xC42.C4 order 128 = 27

Direct product of C2 and C42.C4

G = C₂xC₄².C₄ order 128 = 2⁷

Direct product of C₂ and C₄².C₄