C2^3:2Q8 - GroupNames

G = C₂³⋊₂Q₈ order 64 = 2⁶

2^nd semidirect product of C₂³ and Q₈ acting via Q₈/C₂=C₂²

p-group, metabelian, nilpotent (class 2), monomial, rational

Aliases: C₂³⋊₂Q₈, C₂⁴.₁₈C₂², C₂³.₄₀C₂³, C₂².₃₇C₂⁴, C₂.₁₀2₊¹⁺⁴, C₄⋊C₄⋊₄C₂², C₂²⋊Q₈⋊₁₀C₂, (C₂×Q₈)⋊₄C₂², C₂².₄(C₂×Q₈), C₂.₇(C₂²×Q₈), (C₂×C₄).₂₄C₂³, C₂²⋊C₄.₁₈C₂², (C₂²×C₄).₆₆C₂², (C₂×C₂²⋊C₄).₁₃C₂, SmallGroup(64,224)

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

Derived series

C₁ — C₂² — C₂³⋊₂Q₈

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₂²⋊C₄ — C₂³⋊₂Q₈

Lower central

C₁ — C₂² — C₂³⋊₂Q₈

Upper central

C₁ — C₂² — C₂³⋊₂Q₈

Jennings

C₁ — C₂² — C₂³⋊₂Q₈

Generators and relations for C₂³⋊₂Q₈
G = < a,b,c,d,e | a²=b²=c²=d⁴=1, e²=d², ab=ba, dad^-1=eae^-1=ac=ca, ebe^-1=bc=cb, bd=db, cd=dc, ce=ec, ede^-1=d^-1 >

Subgroups: 193 in 121 conjugacy classes, 81 normal (5 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×Q₈, C₂⁴, C₂×C₂²⋊C₄, C₂²⋊Q₈, C₂³⋊₂Q₈
Quotients: C₁, C₂, C₂², Q₈, C₂³, C₂×Q₈, C₂⁴, C₂²×Q₈, 2₊¹⁺⁴, C₂³⋊₂Q₈

Character table of C₂³⋊₂Q₈

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L
size	1	1	1	1	2	2	2	2	2	2	4	4	4	4	4	4	4	4	4	4	4	4
ρ₁	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	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	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	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	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	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	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	linear of order 2
ρ₉	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
ρ₁₀	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
ρ₁₁	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
ρ₁₂	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
ρ₁₃	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
ρ₁₄	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
ρ₁₅	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
ρ₁₆	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
ρ₁₇	2	-2	2	-2	-2	-2	2	2	2	-2	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from Q₈, Schur index 2
ρ₁₈	2	-2	2	-2	2	-2	-2	-2	2	2	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from Q₈, Schur index 2
ρ₁₉	2	-2	2	-2	2	2	-2	2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from Q₈, Schur index 2
ρ₂₀	2	-2	2	-2	-2	2	2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from Q₈, Schur index 2
ρ₂₁	4	4	-4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from 2₊¹⁺⁴
ρ₂₂	4	-4	-4	4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from 2₊¹⁺⁴

Permutation representations of C₂³⋊₂Q₈
►On 16 points - transitive group 16T97

Generators in S₁₆

(2 11)(4 9)(5 14)(7 16)

(5 14)(6 15)(7 16)(8 13)

(1 10)(2 11)(3 12)(4 9)(5 14)(6 15)(7 16)(8 13)

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

(1 14 3 16)(2 13 4 15)(5 12 7 10)(6 11 8 9)

G:=sub<Sym(16)| (2,11)(4,9)(5,14)(7,16), (5,14)(6,15)(7,16)(8,13), (1,10)(2,11)(3,12)(4,9)(5,14)(6,15)(7,16)(8,13), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,14,3,16)(2,13,4,15)(5,12,7,10)(6,11,8,9)>;

G:=Group( (2,11)(4,9)(5,14)(7,16), (5,14)(6,15)(7,16)(8,13), (1,10)(2,11)(3,12)(4,9)(5,14)(6,15)(7,16)(8,13), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,14,3,16)(2,13,4,15)(5,12,7,10)(6,11,8,9) );

G=PermutationGroup([[(2,11),(4,9),(5,14),(7,16)], [(5,14),(6,15),(7,16),(8,13)], [(1,10),(2,11),(3,12),(4,9),(5,14),(6,15),(7,16),(8,13)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,14,3,16),(2,13,4,15),(5,12,7,10),(6,11,8,9)]])

G:=TransitiveGroup(16,97);

C₂³⋊₂Q₈ is a maximal subgroup of
C₂³⋊SD₁₆ C₂³⋊Q₁₆ C₂⁴.₁₄D₄ C₂⁴.₁₇D₄ C₂⁴⋊₂Q₈ C₂⁴.₁₈₀C₂³ C₂⁴⋊Q₈ C₂⁴.₁₈₂C₂³ C₂².₄₇C₂⁵ C₂².₇₉C₂⁵ C₂².₈₀C₂⁵ C₂².₁₂₄C₂⁵ C₂².₁₅₃C₂⁵ C₂⁴.₇A₄
C_2p.2₊¹⁺⁴: C₂².₄₈C₂⁵ C₂².₉₀C₂⁵ C₂².₁₂₅C₂⁵ C₂².₁₅₀C₂⁵ C₂³⋊₃Dic₆ C₆.₅₁2₊¹⁺⁴ C₂³⋊₂Dic₁₀ C₁₀.₅₁2₊¹⁺⁴ ...
C₂³⋊₂Q₈ is a maximal quotient of
C₂³.₂₁₁C₂⁴ C₂⁴⋊₄Q₈ C₂³.₄₄₉C₂⁴ C₂⁴.₅₈₄C₂³ C₂⁴.₃₅₅C₂³ C₂³.₅₀₈C₂⁴ C₂⁴⋊₅Q₈ C₂³.₅₄₆C₂⁴ C₂³.₅₅₉C₂⁴ C₂⁴.₃₇₉C₂³ C₂³.₅₆₇C₂⁴ C₂³.₆₃₂C₂⁴ C₂⁴.₄₃₄C₂³ C₂⁴.₄₄₈C₂³ C₂⁴.₄₅₀C₂³ C₂³.₆₉₉C₂⁴ C₂⁴.₄₅₆C₂³ C₂³.₇₀₆C₂⁴ C₂³.₇₀₉C₂⁴ C₂³.₇₁₁C₂⁴ C₂⁴⋊₆Q₈ C₂³.₇₄₁C₂⁴ C₂⁴.₁₅Q₈
C₂⁴.D_2p: C₂⁴.₉₁D₄ C₂³⋊₃Dic₆ C₂³⋊₂Dic₁₀ C₂³⋊₂Dic₁₄ ...
C₄⋊C₄⋊D_2p: C₂³.₅₈₃C₂⁴ C₂³.₅₉₂C₂⁴ C₆.₅₁2₊¹⁺⁴ C₁₀.₅₁2₊¹⁺⁴ C₁₄.₅₁2₊¹⁺⁴ ...

Matrix representation of C₂³⋊₂Q₈ ►in GL₆(𝔽₅)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	4	0	0
0	0	0	0	4	0
0	0	0	0	0	1

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	4	0
0	0	0	0	0	4

1	0	0	0	0	0
0	1	0	0	0	0
0	0	4	0	0	0
0	0	0	4	0	0
0	0	0	0	4	0
0	0	0	0	0	4

0	4	0	0	0	0
1	0	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	3	0	0	0	0
3	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(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,1,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,3,0,0,0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

C₂³⋊₂Q₈ in GAP, Magma, Sage, TeX

C_2^3\rtimes_2Q_8

% in TeX

G:=Group("C2^3:2Q8");

// GroupNames label

G:=SmallGroup(64,224);

// by ID

G=gap.SmallGroup(64,224);

# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,192,217,650,188,158,579]);

// Polycyclic

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

// generators/relations

Export

Character table of C₂³⋊₂Q₈ in TeX

G = C23⋊2Q8 order 64 = 26

2nd semidirect product of C23 and Q8 acting via Q8/C2=C22

G = C₂³⋊₂Q₈ order 64 = 2⁶

2^nd semidirect product of C₂³ and Q₈ acting via Q₈/C₂=C₂²