C2^3.Q8 - GroupNames

G = C₂³.Q₈ order 64 = 2⁶

3^rd non-split extension by C₂³ of Q₈ acting via Q₈/C₂=C₂²

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

Aliases: C₂³.₃Q₈, C₂⁴.₇C₂², C₂³.₇₉C₂³, (C₂×C₄).₁₆D₄, C₂.₈(C₄⋊D₄), C₂².₇₂(C₂×D₄), C₂.₈(C₂²⋊Q₈), C₂².₂₂(C₂×Q₈), C₂.C₄²⋊₄C₂, (C₂²×C₄).₈C₂², C₂.₄(C₄²⋊₂C₂), C₂².₃₉(C₄○D₄), (C₂×C₄⋊C₄)⋊₆C₂, (C₂×C₂²⋊C₄).₈C₂, SmallGroup(64,77)

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

Derived series

C₁ — C₂³ — C₂³.Q₈

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂×C₂²⋊C₄ — C₂³.Q₈

Lower central

C₁ — C₂³ — C₂³.Q₈

Upper central

C₁ — C₂³ — C₂³.Q₈

Jennings

C₁ — C₂³ — C₂³.Q₈

Generators and relations for C₂³.Q₈
G = < a,b,c,d,e | a²=b²=c²=d⁴=1, e²=bd², eae^-1=ab=ba, dad^-1=ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d^-1 >

Subgroups: 165 in 93 conjugacy classes, 39 normal (7 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂³.Q₈
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, C₂×D₄, C₂×Q₈, C₄○D₄, C₄⋊D₄, C₂²⋊Q₈, C₄²⋊₂C₂, C₂³.Q₈

Character table of C₂³.Q₈

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	1	1	1	1	4	4	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	2	2	-2	-2	2	2	-2	-2	0	0	0	0	0	0	0	-2	2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	-2	-2	2	-2	2	2	-2	0	0	0	0	0	0	2	0	0	0	0	0	0	-2	orthogonal lifted from D₄
ρ₁₁	2	2	-2	-2	2	2	-2	-2	0	0	0	0	0	0	0	2	-2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	-2	-2	2	-2	2	2	-2	0	0	0	0	0	0	-2	0	0	0	0	0	0	2	orthogonal lifted from D₄
ρ₁₃	2	2	2	-2	-2	-2	2	-2	0	0	0	-2	0	0	0	0	0	0	2	0	0	0	orthogonal lifted from D₄
ρ₁₄	2	2	2	-2	-2	-2	2	-2	0	0	0	2	0	0	0	0	0	0	-2	0	0	0	orthogonal lifted from D₄
ρ₁₅	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
ρ₁₆	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
ρ₁₇	2	-2	-2	-2	2	-2	2	2	0	0	2i	0	0	0	0	0	0	-2i	0	0	0	0	complex lifted from C₄○D₄
ρ₁₈	2	-2	2	-2	-2	2	-2	2	0	0	0	0	-2i	0	0	0	0	0	0	2i	0	0	complex lifted from C₄○D₄
ρ₁₉	2	2	-2	2	-2	-2	-2	2	0	0	0	0	0	2i	0	0	0	0	0	0	-2i	0	complex lifted from C₄○D₄
ρ₂₀	2	2	-2	2	-2	-2	-2	2	0	0	0	0	0	-2i	0	0	0	0	0	0	2i	0	complex lifted from C₄○D₄
ρ₂₁	2	-2	2	-2	-2	2	-2	2	0	0	0	0	2i	0	0	0	0	0	0	-2i	0	0	complex lifted from C₄○D₄
ρ₂₂	2	-2	-2	-2	2	-2	2	2	0	0	-2i	0	0	0	0	0	0	2i	0	0	0	0	complex lifted from C₄○D₄

Smallest permutation representation of C₂³.Q₈
►On 32 points

Generators in S₃₂

(2 26)(4 28)(5 32)(6 16)(7 30)(8 14)(9 15)(10 29)(11 13)(12 31)(18 24)(20 22)

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

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

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

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

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

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

Matrix representation of C₂³.Q₈ ►in GL₆(𝔽₅)

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

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

3	0	0	0	0	0
0	2	0	0	0	0
0	0	0	4	0	0
0	0	4	0	0	0
0	0	0	0	0	2
0	0	0	0	2	0

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[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],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,2,0,0,0,0,2,0] >;

C₂³.Q₈ in GAP, Magma, Sage, TeX

C_2^3.Q_8

% in TeX

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

// GroupNames label

G:=SmallGroup(64,77);

// by ID

G=gap.SmallGroup(64,77);

# by ID

G:=PCGroup([6,-2,2,2,-2,2,2,144,121,55,362,332]);

// Polycyclic

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

// generators/relations

Export

Character table of C₂³.Q₈ in TeX

G = C23.Q8 order 64 = 26

3rd non-split extension by C23 of Q8 acting via Q8/C2=C22

G = C₂³.Q₈ order 64 = 2⁶

3^rd non-split extension by C₂³ of Q₈ acting via Q₈/C₂=C₂²