C9:S4 - GroupNames

G = C₉⋊S₄ order 216 = 2³·3³

The semidirect product of C₉ and S₄ acting via S₄/A₄=C₂

Aliases: C₉⋊S₄, A₄⋊D₉, (C₉×A₄)⋊₂C₂, (C₂×C₁₈)⋊₂S₃, C₃.A₄⋊₁S₃, C₃.₁(C₃⋊S₄), (C₃×A₄).₂S₃, C₂²⋊₁(C₉⋊S₃), (C₂×C₆).₂(C₃⋊S₃), SmallGroup(216,93)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₉×A₄ — C₉⋊S₄

Chief series

C₁ — C₂² — C₂×C₆ — C₂×C₁₈ — C₉×A₄ — C₉⋊S₄

Lower central

C₉×A₄ — C₉⋊S₄

Upper central

C₁

Generators and relations for C₉⋊S₄
G = < a,b,c,d,e | a⁹=b²=c²=d³=e²=1, ab=ba, ac=ca, ad=da, eae=a^-1, dbd^-1=ebe=bc=cb, dcd^-1=b, ce=ec, ede=d^-1 >

C₁

₃C₂

₅₄C₂

C₃

₄C₃

C₂²

₂₇C₄

₂₇C₂²

₃C₆

₁₈S₃

₃₆S₃

C₉

₄C₉

₄C₃²

₂₇D₄

A₄

C₂×C₆

₉Dic₃

₉D₆

₃C₁₈

₆D₉

₁₂C₃⋊S₃

₁₂D₉

₄C₃×C₉

₉C₃⋊D₄

₉S₄

C₂×C₁₈

C₃.A₄

C₃×A₄

C₃.A₄

₃D₁₈

₃Dic₉

₄C₉⋊S₃

₃C₃.S₄

₃C₉⋊D₄

₃C₃⋊S₄

₃C₃.S₄

C₉×A₄

C₉⋊S₄

Character table of C₉⋊S₄

class	1	2A	2B	3A	3B	3C	3D	4	6	9A	9B	9C	9D	9E	9F	9G	9H	9I	18A	18B	18C
size	1	3	54	2	8	8	8	54	6	2	2	2	8	8	8	8	8	8	6	6	6
ρ₁	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	linear of order 2
ρ₃	2	2	0	2	-1	-1	-1	0	2	-1	-1	-1	2	-1	-1	-1	2	2	-1	-1	-1	orthogonal lifted from S₃
ρ₄	2	2	0	2	-1	-1	-1	0	2	2	2	2	-1	-1	-1	-1	-1	-1	2	2	2	orthogonal lifted from S₃
ρ₅	2	2	0	2	-1	-1	-1	0	2	-1	-1	-1	-1	2	2	2	-1	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₆	2	2	0	2	2	2	2	0	2	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₇	2	2	0	-1	-1	-1	2	0	-1	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁷+ζ₉²	ζ₉⁸+ζ₉	ζ₉⁵+ζ₉⁴	ζ₉⁷+ζ₉²	ζ₉⁸+ζ₉	ζ₉⁵+ζ₉⁴	ζ₉⁷+ζ₉²	ζ₉⁸+ζ₉	ζ₉⁵+ζ₉⁴	orthogonal lifted from D₉
ρ₈	2	2	0	-1	2	-1	-1	0	-1	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁵+ζ₉⁴	ζ₉⁵+ζ₉⁴	ζ₉⁷+ζ₉²	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁸+ζ₉	ζ₉⁵+ζ₉⁴	orthogonal lifted from D₉
ρ₉	2	2	0	-1	2	-1	-1	0	-1	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁸+ζ₉	ζ₉⁸+ζ₉	ζ₉⁵+ζ₉⁴	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁷+ζ₉²	ζ₉⁸+ζ₉	orthogonal lifted from D₉
ρ₁₀	2	2	0	-1	-1	-1	2	0	-1	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁵+ζ₉⁴	ζ₉⁷+ζ₉²	ζ₉⁸+ζ₉	ζ₉⁵+ζ₉⁴	ζ₉⁷+ζ₉²	ζ₉⁸+ζ₉	ζ₉⁵+ζ₉⁴	ζ₉⁷+ζ₉²	ζ₉⁸+ζ₉	orthogonal lifted from D₉
ρ₁₁	2	2	0	-1	-1	2	-1	0	-1	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁸+ζ₉	ζ₉⁵+ζ₉⁴	ζ₉⁵+ζ₉⁴	ζ₉⁷+ζ₉²	ζ₉⁷+ζ₉²	ζ₉⁸+ζ₉	ζ₉⁵+ζ₉⁴	orthogonal lifted from D₉
ρ₁₂	2	2	0	-1	-1	-1	2	0	-1	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁸+ζ₉	ζ₉⁵+ζ₉⁴	ζ₉⁷+ζ₉²	ζ₉⁸+ζ₉	ζ₉⁵+ζ₉⁴	ζ₉⁷+ζ₉²	ζ₉⁸+ζ₉	ζ₉⁵+ζ₉⁴	ζ₉⁷+ζ₉²	orthogonal lifted from D₉
ρ₁₃	2	2	0	-1	2	-1	-1	0	-1	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁷+ζ₉²	ζ₉⁷+ζ₉²	ζ₉⁸+ζ₉	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁵+ζ₉⁴	ζ₉⁷+ζ₉²	orthogonal lifted from D₉
ρ₁₄	2	2	0	-1	-1	2	-1	0	-1	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁵+ζ₉⁴	ζ₉⁷+ζ₉²	ζ₉⁷+ζ₉²	ζ₉⁸+ζ₉	ζ₉⁸+ζ₉	ζ₉⁵+ζ₉⁴	ζ₉⁷+ζ₉²	orthogonal lifted from D₉
ρ₁₅	2	2	0	-1	-1	2	-1	0	-1	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁷+ζ₉²	ζ₉⁸+ζ₉	ζ₉⁸+ζ₉	ζ₉⁵+ζ₉⁴	ζ₉⁵+ζ₉⁴	ζ₉⁷+ζ₉²	ζ₉⁸+ζ₉	orthogonal lifted from D₉
ρ₁₆	3	-1	1	3	0	0	0	-1	-1	3	3	3	0	0	0	0	0	0	-1	-1	-1	orthogonal lifted from S₄
ρ₁₇	3	-1	-1	3	0	0	0	1	-1	3	3	3	0	0	0	0	0	0	-1	-1	-1	orthogonal lifted from S₄
ρ₁₈	6	-2	0	6	0	0	0	0	-2	-3	-3	-3	0	0	0	0	0	0	1	1	1	orthogonal lifted from C₃⋊S₄
ρ₁₉	6	-2	0	-3	0	0	0	0	1	3ζ₉⁷+3ζ₉²	3ζ₉⁵+3ζ₉⁴	3ζ₉⁸+3ζ₉	0	0	0	0	0	0	-ζ₉⁵-ζ₉⁴	-ζ₉⁷-ζ₉²	-ζ₉⁸-ζ₉	orthogonal faithful
ρ₂₀	6	-2	0	-3	0	0	0	0	1	3ζ₉⁸+3ζ₉	3ζ₉⁷+3ζ₉²	3ζ₉⁵+3ζ₉⁴	0	0	0	0	0	0	-ζ₉⁷-ζ₉²	-ζ₉⁸-ζ₉	-ζ₉⁵-ζ₉⁴	orthogonal faithful
ρ₂₁	6	-2	0	-3	0	0	0	0	1	3ζ₉⁵+3ζ₉⁴	3ζ₉⁸+3ζ₉	3ζ₉⁷+3ζ₉²	0	0	0	0	0	0	-ζ₉⁸-ζ₉	-ζ₉⁵-ζ₉⁴	-ζ₉⁷-ζ₉²	orthogonal faithful

Smallest permutation representation of C₉⋊S₄
►On 36 points

Generators in S₃₆

(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 33 34 35 36)

(1 12)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 10)(9 11)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)(25 28)(26 29)(27 30)

(1 27)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)

(1 7 4)(2 8 5)(3 9 6)(10 22 31)(11 23 32)(12 24 33)(13 25 34)(14 26 35)(15 27 36)(16 19 28)(17 20 29)(18 21 30)

(2 9)(3 8)(4 7)(5 6)(10 32)(11 31)(12 30)(13 29)(14 28)(15 36)(16 35)(17 34)(18 33)(19 26)(20 25)(21 24)(22 23)

G:=sub<Sym(36)| (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)(28,29,30,31,32,33,34,35,36), (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,10)(9,11)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(25,28)(26,29)(27,30), (1,27)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36), (1,7,4)(2,8,5)(3,9,6)(10,22,31)(11,23,32)(12,24,33)(13,25,34)(14,26,35)(15,27,36)(16,19,28)(17,20,29)(18,21,30), (2,9)(3,8)(4,7)(5,6)(10,32)(11,31)(12,30)(13,29)(14,28)(15,36)(16,35)(17,34)(18,33)(19,26)(20,25)(21,24)(22,23)>;

G:=Group( (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)(28,29,30,31,32,33,34,35,36), (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,10)(9,11)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(25,28)(26,29)(27,30), (1,27)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36), (1,7,4)(2,8,5)(3,9,6)(10,22,31)(11,23,32)(12,24,33)(13,25,34)(14,26,35)(15,27,36)(16,19,28)(17,20,29)(18,21,30), (2,9)(3,8)(4,7)(5,6)(10,32)(11,31)(12,30)(13,29)(14,28)(15,36)(16,35)(17,34)(18,33)(19,26)(20,25)(21,24)(22,23) );

G=PermutationGroup([[(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),(28,29,30,31,32,33,34,35,36)], [(1,12),(2,13),(3,14),(4,15),(5,16),(6,17),(7,18),(8,10),(9,11),(19,31),(20,32),(21,33),(22,34),(23,35),(24,36),(25,28),(26,29),(27,30)], [(1,27),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,25),(9,26),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36)], [(1,7,4),(2,8,5),(3,9,6),(10,22,31),(11,23,32),(12,24,33),(13,25,34),(14,26,35),(15,27,36),(16,19,28),(17,20,29),(18,21,30)], [(2,9),(3,8),(4,7),(5,6),(10,32),(11,31),(12,30),(13,29),(14,28),(15,36),(16,35),(17,34),(18,33),(19,26),(20,25),(21,24),(22,23)]])

C₉⋊S₄ is a maximal subgroup of D₉×S₄
C₉⋊S₄ is a maximal quotient of C₁₈.₅S₄ C₁₈.₆S₄ A₄⋊Dic₉

Matrix representation of C₉⋊S₄ ►in GL₅(𝔽₃₇)

31	20	0	0	0
17	11	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

1	0	0	0	0
0	1	0	0	0
0	0	0	0	1
0	0	36	36	36
0	0	1	0	0

1	0	0	0	0
0	1	0	0	0
0	0	36	36	36
0	0	0	0	1
0	0	0	1	0

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	36	36	36
0	0	0	1	0

1	0	0	0	0
36	36	0	0	0
0	0	1	0	0
0	0	0	0	1
0	0	0	1	0

G:=sub<GL(5,GF(37))| [31,17,0,0,0,20,11,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,36,1,0,0,0,36,0,0,0,1,36,0],[1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,36,0,1,0,0,36,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,1,36,0,0,0,0,36,1,0,0,0,36,0],[1,36,0,0,0,0,36,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

C₉⋊S₄ in GAP, Magma, Sage, TeX

C_9\rtimes S_4

% in TeX

G:=Group("C9:S4");

// GroupNames label

G:=SmallGroup(216,93);

// by ID

G=gap.SmallGroup(216,93);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-2,2,49,542,122,867,3244,1630,1949,2927]);

// Polycyclic

G:=Group<a,b,c,d,e|a^9=b^2=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,d*b*d^-1=e*b*e=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e=d^-1>;

// generators/relations

Export

Subgroup lattice of C₉⋊S₄ in TeX
Character table of C₉⋊S₄ in TeX

G = C9⋊S4 order 216 = 23·33

The semidirect product of C9 and S4 acting via S4/A4=C2

G = C₉⋊S₄ order 216 = 2³·3³

The semidirect product of C₉ and S₄ acting via S₄/A₄=C₂