C3:S3.2D8 - GroupNames

G = C₃⋊S₃.₂D₈ order 288 = 2⁵·3²

1^st non-split extension by C₃⋊S₃ of D₈ acting via D₈/C₄=C₂²

Aliases: C₃⋊S₃.₂D₈, C₄.₁₁S₃≀C₂, (C₃×C₁₂).₃D₄, D₆⋊S₃⋊₁C₄, C₃⋊S₃.₂SD₁₆, D₆⋊D₆.₂C₂, C₁₂.₂₉D₆⋊₇C₂, C₃²⋊₂(D₄⋊C₄), C₂.₅(S₃²⋊C₄), (C₂×C₃⋊S₃).₅D₄, C₄⋊(C₃²⋊C₄)⋊₁C₂, (C₄×C₃⋊S₃).₃C₂², C₃⋊Dic₃.₅(C₂×C₄), (C₃×C₆).₄(C₂²⋊C₄), SmallGroup(288,377)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₃⋊Dic₃ — C₃⋊S₃.₂D₈

Chief series

C₁ — C₃² — C₃×C₆ — C₃⋊Dic₃ — C₄×C₃⋊S₃ — D₆⋊D₆ — C₃⋊S₃.₂D₈

Lower central

C₃² — C₃×C₆ — C₃⋊Dic₃ — C₃⋊S₃.₂D₈

Upper central

C₁ — C₂ — C₄

Generators and relations for C₃⋊S₃.₂D₈
G = < a,b,c,d,e | a³=b³=c²=d⁸=1, e²=c, ab=ba, cac=dbd^-1=a^-1, dad^-1=eae^-1=cbc=b^-1, ebe^-1=a, cd=dc, ce=ec, ede^-1=cd^-1 >

Subgroups: 560 in 88 conjugacy classes, 17 normal (15 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², S₃, C₆, C₈, C₂×C₄, D₄, C₂³, C₃², Dic₃, C₁₂, D₆, C₂×C₆, C₄⋊C₄, C₂×C₈, C₂×D₄, C₃×S₃, C₃⋊S₃, C₃×C₆, C₃⋊C₈, C₂₄, C₄×S₃, D₁₂, C₃⋊D₄, C₃×D₄, C₂²×S₃, D₄⋊C₄, C₃⋊Dic₃, C₃×C₁₂, C₃²⋊C₄, S₃², S₃×C₆, C₂×C₃⋊S₃, S₃×C₈, S₃×D₄, C₃×C₃⋊C₈, D₆⋊S₃, C₃×D₁₂, C₄×C₃⋊S₃, C₂×C₃²⋊C₄, C₂×S₃², C₁₂.₂₉D₆, C₄⋊(C₃²⋊C₄), D₆⋊D₆, C₃⋊S₃.₂D₈
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂²⋊C₄, D₈, SD₁₆, D₄⋊C₄, S₃≀C₂, S₃²⋊C₄, C₃⋊S₃.₂D₈

Character table of C₃⋊S₃.₂D₈

class	1	2A	2B	2C	2D	2E	3A	3B	4A	4B	4C	4D	6A	6B	6C	6D	8A	8B	8C	8D	12A	12B	12C	24A	24B	24C	24D
size	1	1	9	9	12	12	4	4	2	18	36	36	4	4	24	24	6	6	6	6	4	4	8	12	12	12	12
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	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	-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	-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	1	linear of order 2
ρ₅	1	1	-1	-1	1	-1	1	1	-1	1	-i	i	1	1	1	-1	i	-i	i	-i	-1	-1	-1	-i	-i	i	i	linear of order 4
ρ₆	1	1	-1	-1	-1	1	1	1	-1	1	-i	i	1	1	-1	1	-i	i	-i	i	-1	-1	-1	i	i	-i	-i	linear of order 4
ρ₇	1	1	-1	-1	1	-1	1	1	-1	1	i	-i	1	1	1	-1	-i	i	-i	i	-1	-1	-1	i	i	-i	-i	linear of order 4
ρ₈	1	1	-1	-1	-1	1	1	1	-1	1	i	-i	1	1	-1	1	i	-i	i	-i	-1	-1	-1	-i	-i	i	i	linear of order 4
ρ₉	2	2	-2	-2	0	0	2	2	2	-2	0	0	2	2	0	0	0	0	0	0	2	2	2	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	0	0	2	2	-2	-2	0	0	2	2	0	0	0	0	0	0	-2	-2	-2	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	-2	2	-2	0	0	2	2	0	0	0	0	-2	-2	0	0	√2	-√2	-√2	√2	0	0	0	-√2	√2	-√2	√2	orthogonal lifted from D₈
ρ₁₂	2	-2	2	-2	0	0	2	2	0	0	0	0	-2	-2	0	0	-√2	√2	√2	-√2	0	0	0	√2	-√2	√2	-√2	orthogonal lifted from D₈
ρ₁₃	2	-2	-2	2	0	0	2	2	0	0	0	0	-2	-2	0	0	√-2	√-2	-√-2	-√-2	0	0	0	√-2	-√-2	-√-2	√-2	complex lifted from SD₁₆
ρ₁₄	2	-2	-2	2	0	0	2	2	0	0	0	0	-2	-2	0	0	-√-2	-√-2	√-2	√-2	0	0	0	-√-2	√-2	√-2	-√-2	complex lifted from SD₁₆
ρ₁₅	4	4	0	0	0	0	1	-2	4	0	0	0	-2	1	0	0	-2	-2	-2	-2	1	1	-2	1	1	1	1	orthogonal lifted from S₃≀C₂
ρ₁₆	4	4	0	0	2	-2	-2	1	-4	0	0	0	1	-2	-1	1	0	0	0	0	2	2	-1	0	0	0	0	orthogonal lifted from S₃²⋊C₄
ρ₁₇	4	4	0	0	2	2	-2	1	4	0	0	0	1	-2	-1	-1	0	0	0	0	-2	-2	1	0	0	0	0	orthogonal lifted from S₃≀C₂
ρ₁₈	4	4	0	0	-2	-2	-2	1	4	0	0	0	1	-2	1	1	0	0	0	0	-2	-2	1	0	0	0	0	orthogonal lifted from S₃≀C₂
ρ₁₉	4	4	0	0	0	0	1	-2	4	0	0	0	-2	1	0	0	2	2	2	2	1	1	-2	-1	-1	-1	-1	orthogonal lifted from S₃≀C₂
ρ₂₀	4	4	0	0	-2	2	-2	1	-4	0	0	0	1	-2	1	-1	0	0	0	0	2	2	-1	0	0	0	0	orthogonal lifted from S₃²⋊C₄
ρ₂₁	4	4	0	0	0	0	1	-2	-4	0	0	0	-2	1	0	0	-2i	2i	-2i	2i	-1	-1	2	-i	-i	i	i	complex lifted from S₃²⋊C₄
ρ₂₂	4	4	0	0	0	0	1	-2	-4	0	0	0	-2	1	0	0	2i	-2i	2i	-2i	-1	-1	2	i	i	-i	-i	complex lifted from S₃²⋊C₄
ρ₂₃	4	-4	0	0	0	0	1	-2	0	0	0	0	2	-1	0	0	2ζ₈⁷	2ζ₈⁵	2ζ₈³	2ζ₈	3i	-3i	0	ζ₈	ζ₈⁵	ζ₈⁷	ζ₈³	complex faithful
ρ₂₄	4	-4	0	0	0	0	1	-2	0	0	0	0	2	-1	0	0	2ζ₈	2ζ₈³	2ζ₈⁵	2ζ₈⁷	-3i	3i	0	ζ₈⁷	ζ₈³	ζ₈	ζ₈⁵	complex faithful
ρ₂₅	4	-4	0	0	0	0	1	-2	0	0	0	0	2	-1	0	0	2ζ₈³	2ζ₈	2ζ₈⁷	2ζ₈⁵	3i	-3i	0	ζ₈⁵	ζ₈	ζ₈³	ζ₈⁷	complex faithful
ρ₂₆	4	-4	0	0	0	0	1	-2	0	0	0	0	2	-1	0	0	2ζ₈⁵	2ζ₈⁷	2ζ₈	2ζ₈³	-3i	3i	0	ζ₈³	ζ₈⁷	ζ₈⁵	ζ₈	complex faithful
ρ₂₇	8	-8	0	0	0	0	-4	2	0	0	0	0	-2	4	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal faithful

Permutation representations of C₃⋊S₃.₂D₈
►On 24 points - transitive group 24T666

Generators in S₂₄

(2 22 10)(4 24 12)(6 18 14)(8 20 16)

(1 9 21)(3 11 23)(5 13 17)(7 15 19)

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

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

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

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

G:=TransitiveGroup(24,666);

►On 24 points - transitive group 24T667

Generators in S₂₄

(1 21 12)(3 23 14)(5 17 16)(7 19 10)

(2 13 22)(4 15 24)(6 9 18)(8 11 20)

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

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

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

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

G:=TransitiveGroup(24,667);

Matrix representation of C₃⋊S₃.₂D₈ ►in GL₆(𝔽₇₃)

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	0	1
0	0	0	0	72	72

1	0	0	0	0	0
0	1	0	0	0	0
0	0	72	72	0	0
0	0	1	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

72	0	0	0	0	0
0	72	0	0	0	0
0	0	1	0	0	0
0	0	72	72	0	0
0	0	0	0	1	0
0	0	0	0	72	72

6	6	0	0	0	0
67	6	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

6	6	0	0	0	0
6	67	0	0	0	0
0	0	0	0	72	0
0	0	0	0	1	1
0	0	72	0	0	0
0	0	0	72	0	0

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72],[6,67,0,0,0,0,6,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[6,6,0,0,0,0,6,67,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,72,1,0,0,0,0,0,1,0,0] >;

C₃⋊S₃.₂D₈ in GAP, Magma, Sage, TeX

C_3\rtimes S_3._2D_8

% in TeX

G:=Group("C3:S3.2D8");

// GroupNames label

G:=SmallGroup(288,377);

// by ID

G=gap.SmallGroup(288,377);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,85,64,422,100,675,80,2693,2028,691,797,2372]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^8=1,e^2=c,a*b=b*a,c*a*c=d*b*d^-1=a^-1,d*a*d^-1=e*a*e^-1=c*b*c=b^-1,e*b*e^-1=a,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^-1>;

// generators/relations

Export

Character table of C₃⋊S₃.₂D₈ in TeX

G = C3⋊S3.2D8 order 288 = 25·32

1st non-split extension by C3⋊S3 of D8 acting via D8/C4=C22

G = C₃⋊S₃.₂D₈ order 288 = 2⁵·3²

1^st non-split extension by C₃⋊S₃ of D₈ acting via D₈/C₄=C₂²