(C2xQ8):2Q8 - GroupNames

G = (C₂×Q₈)⋊₂Q₈ order 128 = 2⁷

2^nd semidirect product of C₂×Q₈ and Q₈ acting via Q₈/C₂=C₂²

p-group, metabelian, nilpotent (class 3), monomial

Aliases: (C₂×Q₈)⋊₂Q₈, C₄⋊C₄.₈₇D₄, (C₂²×C₄).₇₈D₄, C₂³.₅₈₆(C₂×D₄), C₄.₃₄(C₂²⋊Q₈), C₄²⋊₆C₄.₁₃C₂, C₄.₃₅(C₄.₄D₄), C₂.₉(C₂³⋊Q₈), C₂².₂₁₂C₂²≀C₂, C₂.₂₆(D₄.₉D₄), (C₂²×C₄).₇₂₀C₂³, (C₂×C₄²).₃₅₈C₂², C₂³.₃₈D₄.₇C₂, C₂².₃₂(C₂²⋊Q₈), C₂.₂₆(D₄.₁₀D₄), (C₂²×Q₈).₆₂C₂², C₂².C₄².₁₄C₂, C₄²⋊C₂.₅₄C₂², C₂².₂₄(C₄.₄D₄), (C₂×M₄(2)).₂₂₁C₂², C₂³.₄₁C₂³.₄C₂, C₂³.₆₇C₂³.₁₅C₂, (C₂×C₄).₁₆(C₂×Q₈), (C₂×C₄).₁₀₃₄(C₂×D₄), (C₂×C₄).₃₄₁(C₄○D₄), (C₂×C₄⋊C₄).₁₁₇C₂², SmallGroup(128,760)

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

Derived series

C₁ — C₂²×C₄ — (C₂×Q₈)⋊₂Q₈

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂²×Q₈ — C₂³.₆₇C₂³ — (C₂×Q₈)⋊₂Q₈

Lower central

C₁ — C₂ — C₂²×C₄ — (C₂×Q₈)⋊₂Q₈

Upper central

C₁ — C₂² — C₂²×C₄ — (C₂×Q₈)⋊₂Q₈

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — (C₂×Q₈)⋊₂Q₈

Generators and relations for (C₂×Q₈)⋊₂Q₈
G = < a,b,c,d,e | a²=b⁴=d⁴=1, c²=b², e²=d², ab=ba, ac=ca, ad=da, eae^-1=ab², cbc^-1=dbd^-1=b^-1, be=eb, dcd^-1=ab²c, ece^-1=ab^-1c, ede^-1=d^-1 >

Subgroups: 256 in 123 conjugacy classes, 42 normal (18 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, Q₈, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂×Q₈, C₂.C₄², Q₈⋊C₄, C₂×C₄², C₂×C₄⋊C₄, C₄²⋊C₂, C₂²⋊Q₈, C₄².C₂, C₄⋊Q₈, C₂×M₄(2), C₂²×Q₈, C₄²⋊₆C₄, C₂².C₄², C₂³.₆₇C₂³, C₂³.₃₈D₄, C₂³.₄₁C₂³, (C₂×Q₈)⋊₂Q₈
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, C₂×D₄, C₂×Q₈, C₄○D₄, C₂²≀C₂, C₂²⋊Q₈, C₄.₄D₄, C₂³⋊Q₈, D₄.₉D₄, D₄.₁₀D₄, (C₂×Q₈)⋊₂Q₈

Character table of (C₂×Q₈)⋊₂Q₈

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L	4M	4N	4O	4P	8A	8B	8C	8D
size	1	1	1	1	2	2	2	2	2	2	4	4	4	4	8	8	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	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	2	2	2	2	2	2	-2	-2	-2	-2	0	0	0	0	0	0	-2	0	0	0	0	2	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	-2	-2	2	-2	-2	2	0	0	0	0	0	0	0	2	-2	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	-2	-2	-2	2	2	-2	0	0	0	0	-2	0	0	0	0	2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	-2	-2	2	-2	-2	2	0	0	0	0	0	0	0	-2	2	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	2	2	2	-2	-2	-2	-2	0	0	0	0	0	0	2	0	0	0	0	-2	0	0	0	0	orthogonal lifted from D₄
ρ₁₄	2	2	2	2	-2	-2	-2	2	2	-2	0	0	0	0	2	0	0	0	0	-2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₅	2	-2	-2	2	2	-2	-2	-2	2	2	0	0	0	0	0	2	0	0	0	0	-2	0	0	0	0	0	symplectic lifted from Q₈, Schur index 2
ρ₁₆	2	-2	-2	2	2	-2	-2	-2	2	2	0	0	0	0	0	-2	0	0	0	0	2	0	0	0	0	0	symplectic lifted from Q₈, Schur index 2
ρ₁₇	2	-2	-2	2	-2	2	-2	2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	2i	-2i	0	0	complex lifted from C₄○D₄
ρ₁₈	2	-2	-2	2	-2	2	-2	2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	-2i	2i	0	0	complex lifted from C₄○D₄
ρ₁₉	2	-2	-2	2	-2	2	2	-2	2	-2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	-2i	2i	complex lifted from C₄○D₄
ρ₂₀	2	-2	-2	2	2	-2	2	2	-2	-2	-2i	2i	-2i	2i	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₁	2	-2	-2	2	-2	2	2	-2	2	-2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	2i	-2i	complex lifted from C₄○D₄
ρ₂₂	2	-2	-2	2	2	-2	2	2	-2	-2	2i	-2i	2i	-2i	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₃	4	4	-4	-4	0	0	0	0	0	0	2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from D₄.₁₀D₄, Schur index 2
ρ₂₄	4	4	-4	-4	0	0	0	0	0	0	-2	2	2	-2	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from D₄.₁₀D₄, Schur index 2
ρ₂₅	4	-4	4	-4	0	0	0	0	0	0	2i	2i	-2i	-2i	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from D₄.₉D₄
ρ₂₆	4	-4	4	-4	0	0	0	0	0	0	-2i	-2i	2i	2i	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from D₄.₉D₄

Smallest permutation representation of (C₂×Q₈)⋊₂Q₈
►On 32 points

Generators in S₃₂

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

(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)

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

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

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

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

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

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

Matrix representation of (C₂×Q₈)⋊₂Q₈ ►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

1	0	0	0	0	0
0	1	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

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

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

0	13	0	0	0	0
13	0	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	1	0	0	0
0	0	0	1	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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,13],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,13,0,0,0,0,13,0],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

(C₂×Q₈)⋊₂Q₈ in GAP, Magma, Sage, TeX

(C_2\times Q_8)\rtimes_2Q_8

% in TeX

G:=Group("(C2xQ8):2Q8");

// GroupNames label

G:=SmallGroup(128,760);

// by ID

G=gap.SmallGroup(128,760);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,64,422,387,352,2804,1411,718,172,4037]);

// Polycyclic

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

// generators/relations

Export

Character table of (C₂×Q₈)⋊₂Q₈ in TeX

G = (C2×Q8)⋊2Q8 order 128 = 27

2nd semidirect product of C2×Q8 and Q8 acting via Q8/C2=C22

G = (C₂×Q₈)⋊₂Q₈ order 128 = 2⁷

2^nd semidirect product of C₂×Q₈ and Q₈ acting via Q₈/C₂=C₂²