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G = C32⋊S4order 216 = 23·33

2nd semidirect product of C32 and S4 acting via S4/C22=S3

non-abelian, soluble, monomial

Aliases: C322S4, C625S3, (C3×A4)⋊1S3, C3.3(C3⋊S4), C32⋊A42C2, C22⋊(He3⋊C2), (C2×C6).1(C3⋊S3), SmallGroup(216,95)

Series: Derived Chief Lower central Upper central

C1C2×C6C32⋊A4 — C32⋊S4
C1C22C2×C6C62C32⋊A4 — C32⋊S4
C32⋊A4 — C32⋊S4
C1C3

Generators and relations for C32⋊S4
 G = < a,b,c,d | a6=b6=c3=d2=1, ab=ba, cac-1=a4b, dad=a2b3, cbc-1=a3b, dbd=a3b4, dcd=c-1 >

Subgroups: 310 in 57 conjugacy classes, 11 normal (8 characteristic)
C1, C2 [×2], C3, C3 [×4], C4, C22, C22, S3 [×4], C6 [×4], D4, C32, C32 [×3], Dic3, C12, A4 [×3], D6, C2×C6, C2×C6 [×2], C3×S3 [×4], C3×C6, C3⋊D4, C3×D4, S4 [×3], He3, C3×Dic3, C3×A4 [×3], S3×C6, C62, He3⋊C2, C3×C3⋊D4, C3×S4 [×3], C32⋊A4, C32⋊S4
Quotients: C1, C2, S3 [×4], C3⋊S3, S4, He3⋊C2, C3⋊S4, C32⋊S4

Character table of C32⋊S4

 class 12A2B3A3B3C3D3E3F46A6B6C6D6E6F6G12A12B
 size 1318116242424183366618181818
ρ11111111111111111111    trivial
ρ211-1111111-111111-1-1-1-1    linear of order 2
ρ322022-1-1-12022-1-1-10000    orthogonal lifted from S3
ρ422022-1-12-1022-1-1-10000    orthogonal lifted from S3
ρ522022-12-1-1022-1-1-10000    orthogonal lifted from S3
ρ6220222-1-1-10222220000    orthogonal lifted from S3
ρ73-1-13330001-1-1-1-1-1-1-111    orthogonal lifted from S4
ρ83-11333000-1-1-1-1-1-111-1-1    orthogonal lifted from S4
ρ9331-3+3-3/2-3-3-3/200001-3-3-3/2-3+3-3/2000ζ32ζ3ζ32ζ3    complex lifted from He3⋊C2
ρ103-11-3-3-3/2-3+3-3/20000-1ζ65ζ62-1--3-1+-3ζ3ζ32ζ65ζ6    complex faithful
ρ1133-1-3-3-3/2-3+3-3/20000-1-3+3-3/2-3-3-3/2000ζ65ζ6ζ65ζ6    complex lifted from He3⋊C2
ρ123-11-3+3-3/2-3-3-3/20000-1ζ6ζ652-1+-3-1--3ζ32ζ3ζ6ζ65    complex faithful
ρ13331-3-3-3/2-3+3-3/200001-3+3-3/2-3-3-3/2000ζ3ζ32ζ3ζ32    complex lifted from He3⋊C2
ρ1433-1-3+3-3/2-3-3-3/20000-1-3-3-3/2-3+3-3/2000ζ6ζ65ζ6ζ65    complex lifted from He3⋊C2
ρ153-1-1-3+3-3/2-3-3-3/200001ζ6ζ652-1+-3-1--3ζ6ζ65ζ32ζ3    complex faithful
ρ163-1-1-3-3-3/2-3+3-3/200001ζ65ζ62-1--3-1+-3ζ65ζ6ζ3ζ32    complex faithful
ρ176-2066-30000-2-21110000    orthogonal lifted from C3⋊S4
ρ186-20-3-3-3-3+3-3000001--31+-3-21+-31--30000    complex faithful
ρ196-20-3+3-3-3-3-3000001+-31--3-21--31+-30000    complex faithful

Permutation representations of C32⋊S4
On 18 points - transitive group 18T107
Generators in S18
(1 2)(3 4)(5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)
(1 4 5 2 3 6)(7 12 8 10 9 11)(13 17 15)(14 18 16)
(1 14 10)(2 17 7)(3 18 12)(4 15 9)(5 16 11)(6 13 8)
(1 2)(3 4)(5 6)(7 14)(8 16)(9 18)(10 17)(11 13)(12 15)

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

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

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

G:=TransitiveGroup(18,107);

On 18 points - transitive group 18T108
Generators in S18
(1 2)(3 4)(5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)
(1 4 5 2 3 6)(7 12 8 10 9 11)(13 17 15)(14 18 16)
(1 18 12)(2 15 9)(3 16 11)(4 13 8)(5 14 10)(6 17 7)
(7 17)(8 13)(9 15)(10 14)(11 16)(12 18)

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

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

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

G:=TransitiveGroup(18,108);

C32⋊S4 is a maximal subgroup of   C625D6
C32⋊S4 is a maximal quotient of   C322CSU2(𝔽3)  C323GL2(𝔽3)  C626Dic3

Matrix representation of C32⋊S4 in GL3(𝔽7) generated by

006
031
525
,
611
161
220
,
132
062
030
,
600
002
040
G:=sub<GL(3,GF(7))| [0,0,5,0,3,2,6,1,5],[6,1,2,1,6,2,1,1,0],[1,0,0,3,6,3,2,2,0],[6,0,0,0,0,4,0,2,0] >;

C32⋊S4 in GAP, Magma, Sage, TeX

C_3^2\rtimes S_4
% in TeX

G:=Group("C3^2:S4");
// GroupNames label

G:=SmallGroup(216,95);
// by ID

G=gap.SmallGroup(216,95);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-2,2,49,218,224,3244,1630,1949,2927]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=c^3=d^2=1,a*b=b*a,c*a*c^-1=a^4*b,d*a*d=a^2*b^3,c*b*c^-1=a^3*b,d*b*d=a^3*b^4,d*c*d=c^-1>;
// generators/relations

Export

Character table of C32⋊S4 in TeX

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