(C2xD4):2Q8 - GroupNames

G = (C₂×D₄)⋊₂Q₈ order 128 = 2⁷

2^nd semidirect product of C₂×D₄ and Q₈ acting via Q₈/C₂=C₂²

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

Aliases: (C₂×D₄)⋊₂Q₈, C₄⋊C₄.₈₆D₄, (C₂²×C₄).₇₇D₄, C₄²⋊₆C₄⋊₃₀C₂, C₂³.₅₈₅(C₂×D₄), C₄.₃₃(C₂²⋊Q₈), C₂.₃₂(D₄⋊₄D₄), C₄.₃₄(C₄.₄D₄), C₂.₈(C₂³⋊Q₈), C₂².C₄²⋊₂₀C₂, C₂².₂₁₁C₂²≀C₂, C₂.₂₈(D₄.₈D₄), (C₂×C₄²).₃₅₇C₂², (C₂²×C₄).₇₁₉C₂³, C₂³.₃₇D₄.₇C₂, (C₂²×D₄).₇₄C₂², C₂².₃₁(C₂²⋊Q₈), C₄²⋊C₂.₅₃C₂², C₂³.₄₁C₂³⋊₂C₂, C₂².₂₃(C₄.₄D₄), (C₂×M₄(2)).₂₂₀C₂², C₂⁴.₃C₂².₁₅C₂, (C₂×C₄).₁₅(C₂×Q₈), (C₂×C₄).₁₀₃₃(C₂×D₄), (C₂×C₄).₃₄₀(C₄○D₄), (C₂×C₄⋊C₄).₁₁₆C₂², SmallGroup(128,759)

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

Derived series

C₁ — C₂²×C₄ — (C₂×D₄)⋊₂Q₈

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂²×D₄ — C₂⁴.₃C₂² — (C₂×D₄)⋊₂Q₈

Lower central

C₁ — C₂ — C₂²×C₄ — (C₂×D₄)⋊₂Q₈

Upper central

C₁ — C₂² — C₂²×C₄ — (C₂×D₄)⋊₂Q₈

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — (C₂×D₄)⋊₂Q₈

Generators and relations for (C₂×D₄)⋊₂Q₈
G = < a,b,c,d,e | a²=b⁴=c²=d⁴=1, e²=d², ab=ba, ac=ca, ad=da, eae^-1=ab², cbc=dbd^-1=b^-1, be=eb, dcd^-1=ab²c, ece^-1=abc, ede^-1=d^-1 >

Subgroups: 352 in 141 conjugacy classes, 42 normal (18 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂⁴, D₄⋊C₄, C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₂²⋊Q₈, C₄².C₂, C₄⋊Q₈, C₂×M₄(2), C₂²×D₄, C₄²⋊₆C₄, C₂².C₄², C₂⁴.₃C₂², C₂³.₃₇D₄, C₂³.₄₁C₂³, (C₂×D₄)⋊₂Q₈
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, C₂×D₄, C₂×Q₈, C₄○D₄, C₂²≀C₂, C₂²⋊Q₈, C₄.₄D₄, C₂³⋊Q₈, D₄⋊₄D₄, D₄.₈D₄, (C₂×D₄)⋊₂Q₈

Character table of (C₂×D₄)⋊₂Q₈

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

Smallest permutation representation of (C₂×D₄)⋊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 10)(2 9)(3 12)(4 11)(5 31)(6 30)(7 29)(8 32)(13 15)(18 20)(21 24)(22 23)(25 28)(26 27)

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

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

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

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

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

Matrix representation of (C₂×D₄)⋊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

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	16
0	0	0	0	1	0

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

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

(C₂×D₄)⋊₂Q₈ in GAP, Magma, Sage, TeX

(C_2\times D_4)\rtimes_2Q_8

% in TeX

G:=Group("(C2xD4):2Q8");

// GroupNames label

G:=SmallGroup(128,759);

// by ID

G=gap.SmallGroup(128,759);

# by ID

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

// Polycyclic

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

// generators/relations

Export

Character table of (C₂×D₄)⋊₂Q₈ in TeX

G = (C2×D4)⋊2Q8 order 128 = 27

2nd semidirect product of C2×D4 and Q8 acting via Q8/C2=C22

G = (C₂×D₄)⋊₂Q₈ order 128 = 2⁷

2^nd semidirect product of C₂×D₄ and Q₈ acting via Q₈/C₂=C₂²