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G = S3×F9order 432 = 24·33

Direct product of S3 and F9

direct product, metabelian, soluble, monomial, A-group

Aliases: S3×F9, C33⋊(C2×C8), C3⋊F93C2, C31(C2×F9), (S3×C32)⋊C8, C33⋊C2⋊C8, (C3×F9)⋊2C2, C323(S3×C8), C33⋊C4.C4, C32⋊C4.4D6, (S3×C3⋊S3).C4, C3⋊S3.1(C4×S3), (S3×C32⋊C4).2C2, (C3×C32⋊C4).5C22, (C3×C3⋊S3).(C2×C4), SmallGroup(432,736)

Series: Derived Chief Lower central Upper central

C1C33 — S3×F9
C1C3C33C3×C3⋊S3C3×C32⋊C4S3×C32⋊C4 — S3×F9
C33 — S3×F9
C1

Generators and relations for S3×F9
 G = < a,b,c,d,e | a3=b2=c3=d3=e8=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

Subgroups: 568 in 58 conjugacy classes, 18 normal (all characteristic)
C1, C2 [×3], C3, C3 [×2], C4 [×2], C22, S3, S3 [×4], C6 [×2], C8 [×2], C2×C4, C32, C32 [×2], Dic3, C12, D6 [×2], C2×C8, C3×S3 [×2], C3⋊S3, C3⋊S3 [×3], C3×C6, C3⋊C8, C24, C4×S3, C33, C32⋊C4, C32⋊C4, S32, C2×C3⋊S3, S3×C8, S3×C32, C3×C3⋊S3, C33⋊C2, F9, F9, C2×C32⋊C4, C3×C32⋊C4, C33⋊C4, S3×C3⋊S3, C2×F9, C3×F9, C3⋊F9, S3×C32⋊C4, S3×F9
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C8 [×2], C2×C4, D6, C2×C8, C4×S3, S3×C8, F9, C2×F9, S3×F9

Character table of S3×F9

 class 12A2B2C3A3B3C4A4B4C4D6A6B8A8B8C8D8E8F8G8H12A12B24A24B24C24D
 size 1392728169927271824999927272727181818181818
ρ1111111111111111111111111111    trivial
ρ21111111111111-1-1-1-1-1-1-1-111-1-1-1-1    linear of order 2
ρ31-11-111111-1-11-11111-1-1-1-1111111    linear of order 2
ρ41-11-111111-1-11-1-1-1-1-1111111-1-1-1-1    linear of order 2
ρ51-11-1111-1-1111-1i-i-ii-ii-ii-1-1i-i-ii    linear of order 4
ρ61111111-1-1-1-111i-i-iii-ii-i-1-1i-i-ii    linear of order 4
ρ71111111-1-1-1-111-iii-i-ii-ii-1-1-iii-i    linear of order 4
ρ81-11-1111-1-1111-1-iii-ii-ii-i-1-1-iii-i    linear of order 4
ρ91-1-11111i-ii-i-1-1ζ83ζ85ζ8ζ87ζ87ζ85ζ83ζ8i-iζ83ζ85ζ8ζ87    linear of order 8
ρ101-1-11111-ii-ii-1-1ζ8ζ87ζ83ζ85ζ85ζ87ζ8ζ83-iiζ8ζ87ζ83ζ85    linear of order 8
ρ1111-1-1111i-i-ii-11ζ83ζ85ζ8ζ87ζ83ζ8ζ87ζ85i-iζ83ζ85ζ8ζ87    linear of order 8
ρ1211-1-1111-iii-i-11ζ8ζ87ζ83ζ85ζ8ζ83ζ85ζ87-iiζ8ζ87ζ83ζ85    linear of order 8
ρ131-1-11111-ii-ii-1-1ζ85ζ83ζ87ζ8ζ8ζ83ζ85ζ87-iiζ85ζ83ζ87ζ8    linear of order 8
ρ141-1-11111i-ii-i-1-1ζ87ζ8ζ85ζ83ζ83ζ8ζ87ζ85i-iζ87ζ8ζ85ζ83    linear of order 8
ρ1511-1-1111-iii-i-11ζ85ζ83ζ87ζ8ζ85ζ87ζ8ζ83-iiζ85ζ83ζ87ζ8    linear of order 8
ρ1611-1-1111i-i-ii-11ζ87ζ8ζ85ζ83ζ87ζ85ζ83ζ8i-iζ87ζ8ζ85ζ83    linear of order 8
ρ172020-12-12200-1022220000-1-1-1-1-1-1    orthogonal lifted from S3
ρ182020-12-12200-10-2-2-2-20000-1-11111    orthogonal lifted from D6
ρ192020-12-1-2-200-10-2i2i2i-2i000011i-i-ii    complex lifted from C4×S3
ρ202020-12-1-2-200-102i-2i-2i2i000011-iii-i    complex lifted from C4×S3
ρ2120-20-12-12i-2i001083858870000-iiζ87ζ8ζ85ζ83    complex lifted from S3×C8
ρ2220-20-12-1-2i2i001088783850000i-iζ85ζ83ζ87ζ8    complex lifted from S3×C8
ρ2320-20-12-12i-2i001087885830000-iiζ83ζ85ζ8ζ87    complex lifted from S3×C8
ρ2420-20-12-1-2i2i001085838780000i-iζ8ζ87ζ83ζ85    complex lifted from S3×C8
ρ2588008-1-100000-100000000000000    orthogonal lifted from F9
ρ268-8008-1-100000100000000000000    orthogonal lifted from C2×F9
ρ2716000-8-2100000000000000000000    orthogonal faithful

Permutation representations of S3×F9
On 24 points - transitive group 24T1336
Generators in S24
(1 11 22)(2 12 23)(3 13 24)(4 14 17)(5 15 18)(6 16 19)(7 9 20)(8 10 21)
(1 5)(2 6)(3 7)(4 8)(9 24)(10 17)(11 18)(12 19)(13 20)(14 21)(15 22)(16 23)
(2 12 23)(3 13 24)(4 17 14)(6 19 16)(7 20 9)(8 10 21)
(1 11 22)(3 13 24)(4 14 17)(5 18 15)(7 20 9)(8 21 10)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)

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

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

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

G:=TransitiveGroup(24,1336);

On 27 points - transitive group 27T138
Generators in S27
(1 2 3)(4 20 14)(5 21 15)(6 22 16)(7 23 17)(8 24 18)(9 25 19)(10 26 12)(11 27 13)
(2 3)(12 26)(13 27)(14 20)(15 21)(16 22)(17 23)(18 24)(19 25)
(1 5 9)(2 21 25)(3 15 19)(4 11 6)(7 8 10)(12 17 18)(13 16 14)(20 27 22)(23 24 26)
(1 6 10)(2 22 26)(3 16 12)(4 7 5)(8 9 11)(13 18 19)(14 17 15)(20 23 21)(24 25 27)
(4 5 6 7 8 9 10 11)(12 13 14 15 16 17 18 19)(20 21 22 23 24 25 26 27)

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

G:=Group( (1,2,3)(4,20,14)(5,21,15)(6,22,16)(7,23,17)(8,24,18)(9,25,19)(10,26,12)(11,27,13), (2,3)(12,26)(13,27)(14,20)(15,21)(16,22)(17,23)(18,24)(19,25), (1,5,9)(2,21,25)(3,15,19)(4,11,6)(7,8,10)(12,17,18)(13,16,14)(20,27,22)(23,24,26), (1,6,10)(2,22,26)(3,16,12)(4,7,5)(8,9,11)(13,18,19)(14,17,15)(20,23,21)(24,25,27), (4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19)(20,21,22,23,24,25,26,27) );

G=PermutationGroup([(1,2,3),(4,20,14),(5,21,15),(6,22,16),(7,23,17),(8,24,18),(9,25,19),(10,26,12),(11,27,13)], [(2,3),(12,26),(13,27),(14,20),(15,21),(16,22),(17,23),(18,24),(19,25)], [(1,5,9),(2,21,25),(3,15,19),(4,11,6),(7,8,10),(12,17,18),(13,16,14),(20,27,22),(23,24,26)], [(1,6,10),(2,22,26),(3,16,12),(4,7,5),(8,9,11),(13,18,19),(14,17,15),(20,23,21),(24,25,27)], [(4,5,6,7,8,9,10,11),(12,13,14,15,16,17,18,19),(20,21,22,23,24,25,26,27)])

G:=TransitiveGroup(27,138);

Matrix representation of S3×F9 in GL10(ℤ)

-1100000000
-1000000000
0010000000
0001000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
0100000000
1000000000
0010000000
0001000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
1000000000
0100000000
000-1100000
000-1000100
000-1000001
000-1000010
000-1010000
000-1000000
000-1001000
001-1000000
,
1000000000
0100000000
000000-1000
000001-1000
000100-1000
000010-1000
001000-1000
000000-1010
000000-1001
000000-1100
,
-1000000000
0-100000000
0000000100
0000010000
0000001000
0000000010
0001000000
0000000001
0010000000
0000100000

G:=sub<GL(10,Integers())| [-1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0],[-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0] >;

S3×F9 in GAP, Magma, Sage, TeX

S_3\times F_9
% in TeX

G:=Group("S3xF9");
// GroupNames label

G:=SmallGroup(432,736);
// by ID

G=gap.SmallGroup(432,736);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,36,58,1131,718,165,348,691,530,14118]);
// Polycyclic

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

Export

Character table of S3×F9 in TeX

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