(C2xQ8).Q8 - GroupNames

G = (C₂×Q₈).Q₈ order 128 = 2⁷

2^nd non-split extension by C₂×Q₈ of Q₈ acting via Q₈/C₂=C₂²

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

Aliases: (C₂×Q₈).₂Q₈, (C₂×C₄).₂C₄², (C₂×Q₈).₄₀D₄, C₄.₁₀D₄⋊₅C₄, (C₂×C₄²).₁₃C₄, (C₂²×C₄).₃₈D₄, (C₂²×Q₈).₁C₂², C₂².₁₅(C₂³⋊C₄), C₂.₃(C₄².C₄), C₂.₃(C₄².₃C₄), C₂³.₁₆₂(C₂²⋊C₄), C₂.₁₆(C₂³.₉D₄), C₂³.₆₇C₂³.₃C₂, C₂².₅(C₂.C₄²), (C₂×C₄⋊C₄).₁₀C₄, (C₂×C₄).₃(C₄⋊C₄), (C₂×Q₈).₄₁(C₂×C₄), (C₂×C₄).₃(C₂²⋊C₄), (C₂²×C₄).₆₇(C₂×C₄), (C₂×C₄.₁₀D₄).₅C₂, SmallGroup(128,126)

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

Derived series

C₁ — C₂×C₄ — (C₂×Q₈).Q₈

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂²×Q₈ — C₂³.₆₇C₂³ — (C₂×Q₈).Q₈

Lower central

C₁ — C₂ — C₂² — C₂×C₄ — (C₂×Q₈).Q₈

Upper central

C₁ — C₂² — C₂³ — C₂²×Q₈ — (C₂×Q₈).Q₈

Jennings

C₁ — C₂ — C₂² — C₂²×Q₈ — (C₂×Q₈).Q₈

Generators and relations for (C₂×Q₈).Q₈
G = < a,b,c,d,e | a²=b⁴=d⁴=1, c²=b², e²=ab^-1d², dbd^-1=ab=ba, ece^-1=ac=ca, ad=da, eae^-1=ab², cbc^-1=b^-1, be=eb, cd=dc, ede^-1=cd^-1 >

Subgroups: 192 in 86 conjugacy classes, 32 normal (18 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, Q₈, C₂³, C₄², C₄⋊C₄, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂×Q₈, C₂×Q₈, C₂.C₄², C₄.₁₀D₄, C₄.₁₀D₄, C₂×C₄², C₂×C₄⋊C₄, C₂×M₄(2), C₂²×Q₈, C₂³.₆₇C₂³, C₂×C₄.₁₀D₄, (C₂×Q₈).Q₈
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, Q₈, C₄², C₂²⋊C₄, C₄⋊C₄, C₂.C₄², C₂³⋊C₄, C₂³.₉D₄, C₄².C₄, C₄².₃C₄, (C₂×Q₈).Q₈

Character table of (C₂×Q₈).Q₈

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	i	1	-1	-1	1	i	-1	-i	-i	1	i	-i	1	1	-i	-i	i	i	-1	-1	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	-i	1	-1	1	-1	-i	1	i	i	-1	i	-i	-i	i	1	-1	-1	1	i	-i	linear of order 4
ρ₉	1	-1	1	-1	-1	1	i	1	-1	-1	1	i	-1	-i	-i	1	i	-i	-1	-1	i	i	-i	-i	1	1	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	-i	1	-1	1	-1	-i	1	i	i	-1	i	-i	i	-i	-1	1	1	-1	-i	i	linear of order 4
ρ₁₃	1	-1	1	-1	-1	1	-i	1	-1	-1	1	-i	-1	i	i	1	-i	i	-1	-1	-i	-i	i	i	1	1	linear of order 4
ρ₁₄	1	-1	1	-1	-1	1	i	1	-1	1	-1	i	1	-i	-i	-1	-i	i	i	-i	1	-1	-1	1	-i	i	linear of order 4
ρ₁₅	1	-1	1	-1	-1	1	-i	1	-1	-1	1	-i	-1	i	i	1	-i	i	1	1	i	i	-i	-i	-1	-1	linear of order 4
ρ₁₆	1	-1	1	-1	-1	1	i	1	-1	1	-1	i	1	-i	-i	-1	-i	i	-i	i	-1	1	1	-1	i	-i	linear of order 4
ρ₁₇	2	2	2	2	2	2	0	-2	-2	2	2	0	-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	-2	-2	-2	0	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	2	-2	2	0	2	0	0	-2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₂₀	2	-2	2	-2	-2	2	0	-2	2	2	-2	0	-2	0	0	2	0	0	0	0	0	0	0	0	0	0	symplectic lifted from Q₈, Schur index 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	-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	-2	0	0	0	0	2	0	2	-2	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₄².₃C₄, Schur index 2
ρ₂₄	4	4	-4	-4	0	0	2	0	0	0	0	-2	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	0	0	2i	0	0	0	0	-2i	0	2i	-2i	0	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄².C₄
ρ₂₆	4	-4	-4	4	0	0	-2i	0	0	0	0	2i	0	-2i	2i	0	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄².C₄

Smallest permutation representation of (C₂×Q₈).Q₈
►On 32 points

Generators in S₃₂

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

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

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

(2 22 6 18)(3 29)(4 15 8 11)(7 25)(9 32 13 28)(10 23)(14 19)(20 26 24 30)

(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)| (1,27)(2,32)(3,29)(4,26)(5,31)(6,28)(7,25)(8,30)(9,18)(10,23)(11,20)(12,17)(13,22)(14,19)(15,24)(16,21), (1,25,5,29)(2,26,6,30)(3,27,7,31)(4,28,8,32)(9,20,13,24)(10,21,14,17)(11,22,15,18)(12,23,16,19), (1,21,5,17)(2,9,6,13)(3,19,7,23)(4,15,8,11)(10,29,14,25)(12,27,16,31)(18,28,22,32)(20,26,24,30), (2,22,6,18)(3,29)(4,15,8,11)(7,25)(9,32,13,28)(10,23)(14,19)(20,26,24,30), (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( (1,27)(2,32)(3,29)(4,26)(5,31)(6,28)(7,25)(8,30)(9,18)(10,23)(11,20)(12,17)(13,22)(14,19)(15,24)(16,21), (1,25,5,29)(2,26,6,30)(3,27,7,31)(4,28,8,32)(9,20,13,24)(10,21,14,17)(11,22,15,18)(12,23,16,19), (1,21,5,17)(2,9,6,13)(3,19,7,23)(4,15,8,11)(10,29,14,25)(12,27,16,31)(18,28,22,32)(20,26,24,30), (2,22,6,18)(3,29)(4,15,8,11)(7,25)(9,32,13,28)(10,23)(14,19)(20,26,24,30), (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([[(1,27),(2,32),(3,29),(4,26),(5,31),(6,28),(7,25),(8,30),(9,18),(10,23),(11,20),(12,17),(13,22),(14,19),(15,24),(16,21)], [(1,25,5,29),(2,26,6,30),(3,27,7,31),(4,28,8,32),(9,20,13,24),(10,21,14,17),(11,22,15,18),(12,23,16,19)], [(1,21,5,17),(2,9,6,13),(3,19,7,23),(4,15,8,11),(10,29,14,25),(12,27,16,31),(18,28,22,32),(20,26,24,30)], [(2,22,6,18),(3,29),(4,15,8,11),(7,25),(9,32,13,28),(10,23),(14,19),(20,26,24,30)], [(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 (C₂×Q₈).Q₈ ►in GL₆(𝔽₁₇)

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	16	0
0	0	0	0	0	16

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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	16	0	0
0	0	1	0	0	0
0	0	0	0	0	1
0	0	0	0	16	0

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

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

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

(C₂×Q₈).Q₈ in GAP, Magma, Sage, TeX

(C_2\times Q_8).Q_8

% in TeX

G:=Group("(C2xQ8).Q8");

// GroupNames label

G:=SmallGroup(128,126);

// by ID

G=gap.SmallGroup(128,126);

# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,758,352,1466,521,248,2804,4037]);

// Polycyclic

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

// generators/relations

Export

Character table of (C₂×Q₈).Q₈ in TeX

G = (C2×Q8).Q8 order 128 = 27

2nd non-split extension by C2×Q8 of Q8 acting via Q8/C2=C22

G = (C₂×Q₈).Q₈ order 128 = 2⁷

2^nd non-split extension by C₂×Q₈ of Q₈ acting via Q₈/C₂=C₂²