C3:S3.Q8 - GroupNames

G = C₃⋊S₃.Q₈ order 144 = 2⁴·3²

The non-split extension by C₃⋊S₃ of Q₈ acting via Q₈/C₂=C₂²

Aliases: C₃⋊S₃.Q₈, C₂.₂S₃≀C₂, C₃²⋊C₄⋊₁C₄, (C₃×C₆).₂D₄, C₃²⋊₁(C₄⋊C₄), C₆.D₆.₂C₂, C₃⋊S₃.₃(C₂×C₄), (C₂×C₃²⋊C₄).₁C₂, (C₂×C₃⋊S₃).₂C₂², SmallGroup(144,116)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₃⋊S₃ — C₃⋊S₃.Q₈

Chief series

C₁ — C₃² — C₃⋊S₃ — C₂×C₃⋊S₃ — C₆.D₆ — C₃⋊S₃.Q₈

Lower central

C₃² — C₃⋊S₃ — C₃⋊S₃.Q₈

Upper central

C₁ — C₂

Generators and relations for C₃⋊S₃.Q₈
G = < a,b,c,d,e | a³=b³=c²=d⁴=1, e²=d², ab=ba, cac=dbd^-1=a^-1, dad^-1=cbc=ebe^-1=b^-1, ae=ea, cd=dc, ce=ec, ede^-1=cd^-1 >

C₁

C₂

₉C₂

₂C₃

₆C₄

₉C₄

₉C₂²

₉C₄

₂C₆

₆S₃

C₃²

₉C₂×C₄

₂Dic₃

₆C₁₂

₆D₆

C₃⋊S₃

C₃×C₆

₉C₄⋊C₄

₆C₄×S₃

C₃²⋊C₄

C₂×C₃⋊S₃

C₃²⋊C₄

₂C₃×Dic₃

C₆.D₆

C₂×C₃²⋊C₄

C₃⋊S₃.Q₈

Character table of C₃⋊S₃.Q₈

class	1	2A	2B	2C	3A	3B	4A	4B	4C	4D	4E	4F	6A	6B	12A	12B	12C	12D
size	1	1	9	9	4	4	6	6	6	6	18	18	4	4	12	12	12	12
ρ₁	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	linear of order 2
ρ₃	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	linear of order 2
ρ₅	1	-1	-1	1	1	1	-i	i	i	-i	-1	1	-1	-1	i	-i	i	-i	linear of order 4
ρ₆	1	-1	-1	1	1	1	i	i	-i	-i	1	-1	-1	-1	i	i	-i	-i	linear of order 4
ρ₇	1	-1	-1	1	1	1	-i	-i	i	i	1	-1	-1	-1	-i	-i	i	i	linear of order 4
ρ₈	1	-1	-1	1	1	1	i	-i	-i	i	-1	1	-1	-1	-i	i	-i	i	linear of order 4
ρ₉	2	2	-2	-2	2	2	0	0	0	0	0	0	2	2	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	-2	2	-2	2	2	0	0	0	0	0	0	-2	-2	0	0	0	0	symplectic lifted from Q₈, Schur index 2
ρ₁₁	4	4	0	0	1	-2	0	2	0	2	0	0	-2	1	-1	0	0	-1	orthogonal lifted from S₃≀C₂
ρ₁₂	4	4	0	0	-2	1	2	0	2	0	0	0	1	-2	0	-1	-1	0	orthogonal lifted from S₃≀C₂
ρ₁₃	4	4	0	0	-2	1	-2	0	-2	0	0	0	1	-2	0	1	1	0	orthogonal lifted from S₃≀C₂
ρ₁₄	4	4	0	0	1	-2	0	-2	0	-2	0	0	-2	1	1	0	0	1	orthogonal lifted from S₃≀C₂
ρ₁₅	4	-4	0	0	-2	1	-2i	0	2i	0	0	0	-1	2	0	i	-i	0	complex faithful
ρ₁₆	4	-4	0	0	1	-2	0	-2i	0	2i	0	0	2	-1	i	0	0	-i	complex faithful
ρ₁₇	4	-4	0	0	1	-2	0	2i	0	-2i	0	0	2	-1	-i	0	0	i	complex faithful
ρ₁₈	4	-4	0	0	-2	1	2i	0	-2i	0	0	0	-1	2	0	-i	i	0	complex faithful

Permutation representations of C₃⋊S₃.Q₈
►On 24 points - transitive group 24T214

Generators in S₂₄

(1 20 15)(3 18 13)(6 11 24)(8 9 22)

(2 16 17)(4 14 19)(5 23 10)(7 21 12)

(1 3)(2 4)(5 12)(6 9)(7 10)(8 11)(13 20)(14 17)(15 18)(16 19)(21 23)(22 24)

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)

(1 24 3 22)(2 21 4 23)(5 16 7 14)(6 18 8 20)(9 15 11 13)(10 17 12 19)

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

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

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

G:=TransitiveGroup(24,214);

►On 24 points - transitive group 24T216

Generators in S₂₄

(1 13 18)(3 15 20)(6 11 22)(8 9 24)

(2 19 14)(4 17 16)(5 21 10)(7 23 12)

(5 10)(6 11)(7 12)(8 9)(13 18)(14 19)(15 20)(16 17)

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)

(1 24 3 22)(2 23 4 21)(5 19 7 17)(6 13 8 15)(9 20 11 18)(10 14 12 16)

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

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

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

G:=TransitiveGroup(24,216);

C₃⋊S₃.Q₈ is a maximal subgroup of
PSU₃(𝔽₂)⋊C₄ F₉⋊C₄ S₃²⋊Q₈ C₃²⋊C₄⋊Q₈ C₄×S₃≀C₂ C₆².₉D₄ C₆²⋊D₄ C₃³⋊C₄⋊C₄ (C₃×C₆).₉D₁₂
C₃⋊S₃.Q₈ is a maximal quotient of
C₃²⋊C₄⋊C₈ C₄.₁₉S₃≀C₂ C₆².D₄ C₆².₆D₄ C₆².₇D₄ C₆.S₃≀C₂ C₃³⋊C₄⋊C₄ (C₃×C₆).₉D₁₂

Matrix representation of C₃⋊S₃.Q₈ ►in GL₄(𝔽₅) generated by

4	0	1	0
0	0	3	4
4	0	0	0
2	1	3	4

4	2	0	0
2	0	0	0
0	2	0	3
2	1	3	4

4	0	0	2
0	0	2	1
0	3	0	2
0	0	0	1

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

2	1	0	0
0	3	0	0
0	1	0	4
0	3	1	0

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

C₃⋊S₃.Q₈ in GAP, Magma, Sage, TeX

C_3\rtimes S_3.Q_8

% in TeX

G:=Group("C3:S3.Q8");

// GroupNames label

G:=SmallGroup(144,116);

// by ID

G=gap.SmallGroup(144,116);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,3,48,73,55,964,730,256,299,881]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^4=1,e^2=d^2,a*b=b*a,c*a*c=d*b*d^-1=a^-1,d*a*d^-1=c*b*c=e*b*e^-1=b^-1,a*e=e*a,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^-1>;

// generators/relations

Export

Subgroup lattice of C₃⋊S₃.Q₈ in TeX
Character table of C₃⋊S₃.Q₈ in TeX

G = C3⋊S3.Q8 order 144 = 24·32

The non-split extension by C3⋊S3 of Q8 acting via Q8/C2=C22

G = C₃⋊S₃.Q₈ order 144 = 2⁴·3²

The non-split extension by C₃⋊S₃ of Q₈ acting via Q₈/C₂=C₂²