C3:S3.5D8 - GroupNames

G = C₃⋊S₃.₅D₈ order 288 = 2⁵·3²

The non-split extension by C₃⋊S₃ of D₈ acting via D₈/D₄=C₂

Aliases: C₃⋊S₃.₅D₈, C₁₂⋊S₃⋊₁C₄, D₄⋊₁(C₃²⋊C₄), (D₄×C₃²)⋊₁C₄, C₃⋊S₃.₆SD₁₆, C₃⋊Dic₃.₁₀D₄, C₃²⋊₇(D₄⋊C₄), C₂.₆(C₆²⋊C₄), (D₄×C₃⋊S₃).₁C₂, C₃⋊S₃⋊₃C₈⋊₁C₂, C₄.₁(C₂×C₃²⋊C₄), C₄⋊(C₃²⋊C₄)⋊₂C₂, (C₃×C₁₂).₁(C₂×C₄), (C₂×C₃⋊S₃).₅₁D₄, (C₄×C₃⋊S₃).₅C₂², (C₃×C₆).₁₆(C₂²⋊C₄), SmallGroup(288,430)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₁₂ — C₃⋊S₃.₅D₈

Chief series

C₁ — C₃² — C₃×C₆ — C₂×C₃⋊S₃ — C₄×C₃⋊S₃ — C₄⋊(C₃²⋊C₄) — C₃⋊S₃.₅D₈

Lower central

C₃² — C₃×C₆ — C₃×C₁₂ — C₃⋊S₃.₅D₈

Upper central

C₁ — C₂ — C₄ — D₄

Generators and relations for C₃⋊S₃.₅D₈
G = < a,b,c,d,e | a³=b³=c²=d⁸=1, e²=c, ab=ba, cac=a^-1, dad^-1=eae^-1=ab^-1, cbc=b^-1, dbd^-1=ebe^-1=a^-1b^-1, cd=dc, ce=ec, ede^-1=cd^-1 >

Subgroups: 720 in 104 conjugacy classes, 18 normal (all characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², S₃, C₆, C₈, C₂×C₄, D₄, D₄, C₂³, C₃², Dic₃, C₁₂, D₆, C₂×C₆, C₄⋊C₄, C₂×C₈, C₂×D₄, C₃⋊S₃, C₃⋊S₃, C₃×C₆, C₃×C₆, C₄×S₃, D₁₂, C₃⋊D₄, C₃×D₄, C₂²×S₃, D₄⋊C₄, C₃⋊Dic₃, C₃×C₁₂, C₃²⋊C₄, C₂×C₃⋊S₃, C₂×C₃⋊S₃, C₆², S₃×D₄, C₃²⋊₂C₈, C₄×C₃⋊S₃, C₁₂⋊S₃, C₃²⋊₇D₄, D₄×C₃², C₂×C₃²⋊C₄, C₂²×C₃⋊S₃, C₃⋊S₃⋊₃C₈, C₄⋊(C₃²⋊C₄), D₄×C₃⋊S₃, C₃⋊S₃.₅D₈
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂²⋊C₄, D₈, SD₁₆, D₄⋊C₄, C₃²⋊C₄, C₂×C₃²⋊C₄, C₆²⋊C₄, C₃⋊S₃.₅D₈

Character table of C₃⋊S₃.₅D₈

class	1	2A	2B	2C	2D	2E	3A	3B	4A	4B	4C	4D	6A	6B	6C	6D	6E	6F	8A	8B	8C	8D	12A	12B
size	1	1	4	9	9	36	4	4	2	18	36	36	4	4	8	8	8	8	18	18	18	18	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	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	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	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	linear of order 2
ρ₅	1	1	1	-1	-1	-1	1	1	1	-1	i	-i	1	1	1	1	1	1	i	-i	-i	i	1	1	linear of order 4
ρ₆	1	1	-1	-1	-1	1	1	1	1	-1	i	-i	1	1	-1	-1	-1	-1	-i	i	i	-i	1	1	linear of order 4
ρ₇	1	1	-1	-1	-1	1	1	1	1	-1	-i	i	1	1	-1	-1	-1	-1	i	-i	-i	i	1	1	linear of order 4
ρ₈	1	1	1	-1	-1	-1	1	1	1	-1	-i	i	1	1	1	1	1	1	-i	i	i	-i	1	1	linear of order 4
ρ₉	2	2	0	2	2	0	2	2	-2	-2	0	0	2	2	0	0	0	0	0	0	0	0	-2	-2	orthogonal lifted from D₄
ρ₁₀	2	2	0	-2	-2	0	2	2	-2	2	0	0	2	2	0	0	0	0	0	0	0	0	-2	-2	orthogonal lifted from D₄
ρ₁₁	2	-2	0	-2	2	0	2	2	0	0	0	0	-2	-2	0	0	0	0	-√2	√2	-√2	√2	0	0	orthogonal lifted from D₈
ρ₁₂	2	-2	0	-2	2	0	2	2	0	0	0	0	-2	-2	0	0	0	0	√2	-√2	√2	-√2	0	0	orthogonal lifted from D₈
ρ₁₃	2	-2	0	2	-2	0	2	2	0	0	0	0	-2	-2	0	0	0	0	√-2	√-2	-√-2	-√-2	0	0	complex lifted from SD₁₆
ρ₁₄	2	-2	0	2	-2	0	2	2	0	0	0	0	-2	-2	0	0	0	0	-√-2	-√-2	√-2	√-2	0	0	complex lifted from SD₁₆
ρ₁₅	4	4	-4	0	0	0	1	-2	4	0	0	0	-2	1	2	-1	-1	2	0	0	0	0	-2	1	orthogonal lifted from C₂×C₃²⋊C₄
ρ₁₆	4	4	0	0	0	0	-2	1	-4	0	0	0	1	-2	3	0	0	-3	0	0	0	0	-1	2	orthogonal lifted from C₆²⋊C₄
ρ₁₇	4	4	0	0	0	0	-2	1	-4	0	0	0	1	-2	-3	0	0	3	0	0	0	0	-1	2	orthogonal lifted from C₆²⋊C₄
ρ₁₈	4	4	4	0	0	0	1	-2	4	0	0	0	-2	1	-2	1	1	-2	0	0	0	0	-2	1	orthogonal lifted from C₃²⋊C₄
ρ₁₉	4	4	0	0	0	0	1	-2	-4	0	0	0	-2	1	0	3	-3	0	0	0	0	0	2	-1	orthogonal lifted from C₆²⋊C₄
ρ₂₀	4	4	-4	0	0	0	-2	1	4	0	0	0	1	-2	-1	2	2	-1	0	0	0	0	1	-2	orthogonal lifted from C₂×C₃²⋊C₄
ρ₂₁	4	4	4	0	0	0	-2	1	4	0	0	0	1	-2	1	-2	-2	1	0	0	0	0	1	-2	orthogonal lifted from C₃²⋊C₄
ρ₂₂	4	4	0	0	0	0	1	-2	-4	0	0	0	-2	1	0	-3	3	0	0	0	0	0	2	-1	orthogonal lifted from C₆²⋊C₄
ρ₂₃	8	-8	0	0	0	0	2	-4	0	0	0	0	4	-2	0	0	0	0	0	0	0	0	0	0	orthogonal faithful
ρ₂₄	8	-8	0	0	0	0	-4	2	0	0	0	0	-2	4	0	0	0	0	0	0	0	0	0	0	orthogonal faithful

Permutation representations of C₃⋊S₃.₅D₈
►On 24 points - transitive group 24T676

Generators in S₂₄

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

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

(9 23)(10 24)(11 17)(12 18)(13 19)(14 20)(15 21)(16 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)

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

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

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

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

G:=TransitiveGroup(24,676);

►On 24 points - transitive group 24T677

Generators in S₂₄

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

(2 23 14)(4 16 17)(6 19 10)(8 12 21)

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

(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 21 22 16)(10 15 23 20)(11 19 24 14)(12 13 17 18)

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

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

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

G:=TransitiveGroup(24,677);

Matrix representation of C₃⋊S₃.₅D₈ ►in GL₆(𝔽₇₃)

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	0	1
0	0	0	0	72	72

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

61	67	0	0	0	0
12	0	0	0	0	0
0	0	0	0	72	1
0	0	0	0	71	72
0	0	49	49	0	0
0	0	49	48	0	0

0	6	0	0	0	0
12	0	0	0	0	0
0	0	0	0	1	72
0	0	0	0	2	1
0	0	49	49	0	0
0	0	49	48	0	0

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72],[61,12,0,0,0,0,67,0,0,0,0,0,0,0,0,0,49,49,0,0,0,0,49,48,0,0,72,71,0,0,0,0,1,72,0,0],[0,12,0,0,0,0,6,0,0,0,0,0,0,0,0,0,49,49,0,0,0,0,49,48,0,0,1,2,0,0,0,0,72,1,0,0] >;

C₃⋊S₃.₅D₈ in GAP, Magma, Sage, TeX

C_3\rtimes S_3._5D_8

% in TeX

G:=Group("C3:S3.5D8");

// GroupNames label

G:=SmallGroup(288,430);

// by ID

G=gap.SmallGroup(288,430);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,675,346,80,9413,691,12550,2372]);

// 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=a^-1,d*a*d^-1=e*a*e^-1=a*b^-1,c*b*c=b^-1,d*b*d^-1=e*b*e^-1=a^-1*b^-1,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^-1>;

// generators/relations

Export

Character table of C₃⋊S₃.₅D₈ in TeX

G = C3⋊S3.5D8 order 288 = 25·32

The non-split extension by C3⋊S3 of D8 acting via D8/D4=C2

G = C₃⋊S₃.₅D₈ order 288 = 2⁵·3²

The non-split extension by C₃⋊S₃ of D₈ acting via D₈/D₄=C₂