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G = C3×F9order 216 = 23·33

Direct product of C3 and F9

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

Aliases: C3×F9, C32⋊C24, C331C8, C3⋊S3.C12, C32⋊C4.1C6, (C3×C3⋊S3).1C4, (C3×C32⋊C4).1C2, SmallGroup(216,154)

Series: Derived Chief Lower central Upper central

C1C32 — C3×F9
C1C32C3⋊S3C32⋊C4C3×C32⋊C4 — C3×F9
C32 — C3×F9
C1C3

Generators and relations for C3×F9
 G = < a,b,c,d | a3=b3=c3=d8=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

9C2
4C3
8C3
9C4
9C6
12S3
4C32
8C32
9C8
9C12
12C3×S3
9C24

Character table of C3×F9

 class 123A3B3C3D3E4A4B6A6B8A8B8C8D12A12B12C12D24A24B24C24D24E24F24G24H
 size 191188899999999999999999999
ρ1111111111111111111111111111    trivial
ρ211111111111-1-1-1-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ311ζ32ζ31ζ32ζ311ζ3ζ321111ζ3ζ32ζ3ζ32ζ32ζ32ζ32ζ32ζ3ζ3ζ3ζ3    linear of order 3
ρ411ζ3ζ321ζ3ζ3211ζ32ζ3-1-1-1-1ζ32ζ3ζ32ζ3ζ65ζ65ζ65ζ65ζ6ζ6ζ6ζ6    linear of order 6
ρ511ζ3ζ321ζ3ζ3211ζ32ζ31111ζ32ζ3ζ32ζ3ζ3ζ3ζ3ζ3ζ32ζ32ζ32ζ32    linear of order 3
ρ611ζ32ζ31ζ32ζ311ζ3ζ32-1-1-1-1ζ3ζ32ζ3ζ32ζ6ζ6ζ6ζ6ζ65ζ65ζ65ζ65    linear of order 6
ρ71111111-1-111i-i-ii-1-1-1-1-i-iii-i-iii    linear of order 4
ρ81111111-1-111-iii-i-1-1-1-1ii-i-iii-i-i    linear of order 4
ρ91-111111i-i-1-1ζ83ζ85ζ8ζ87i-i-iiζ85ζ8ζ87ζ83ζ85ζ8ζ87ζ83    linear of order 8
ρ101-111111-ii-1-1ζ85ζ83ζ87ζ8-iii-iζ83ζ87ζ8ζ85ζ83ζ87ζ8ζ85    linear of order 8
ρ111-111111i-i-1-1ζ87ζ8ζ85ζ83i-i-iiζ8ζ85ζ83ζ87ζ8ζ85ζ83ζ87    linear of order 8
ρ121-111111-ii-1-1ζ8ζ87ζ83ζ85-iii-iζ87ζ83ζ85ζ8ζ87ζ83ζ85ζ8    linear of order 8
ρ1311ζ3ζ321ζ3ζ32-1-1ζ32ζ3-iii-iζ6ζ65ζ6ζ65ζ4ζ3ζ4ζ3ζ43ζ3ζ43ζ3ζ4ζ32ζ4ζ32ζ43ζ32ζ43ζ32    linear of order 12
ρ1411ζ32ζ31ζ32ζ3-1-1ζ3ζ32i-i-iiζ65ζ6ζ65ζ6ζ43ζ32ζ43ζ32ζ4ζ32ζ4ζ32ζ43ζ3ζ43ζ3ζ4ζ3ζ4ζ3    linear of order 12
ρ1511ζ3ζ321ζ3ζ32-1-1ζ32ζ3i-i-iiζ6ζ65ζ6ζ65ζ43ζ3ζ43ζ3ζ4ζ3ζ4ζ3ζ43ζ32ζ43ζ32ζ4ζ32ζ4ζ32    linear of order 12
ρ1611ζ32ζ31ζ32ζ3-1-1ζ3ζ32-iii-iζ65ζ6ζ65ζ6ζ4ζ32ζ4ζ32ζ43ζ32ζ43ζ32ζ4ζ3ζ4ζ3ζ43ζ3ζ43ζ3    linear of order 12
ρ171-1ζ32ζ31ζ32ζ3-iiζ65ζ6ζ85ζ83ζ87ζ8ζ86ζ3ζ82ζ32ζ82ζ3ζ86ζ32ζ83ζ32ζ87ζ32ζ8ζ32ζ85ζ32ζ83ζ3ζ87ζ3ζ8ζ3ζ85ζ3    linear of order 24
ρ181-1ζ3ζ321ζ3ζ32-iiζ6ζ65ζ85ζ83ζ87ζ8ζ86ζ32ζ82ζ3ζ82ζ32ζ86ζ3ζ83ζ3ζ87ζ3ζ8ζ3ζ85ζ3ζ83ζ32ζ87ζ32ζ8ζ32ζ85ζ32    linear of order 24
ρ191-1ζ3ζ321ζ3ζ32-iiζ6ζ65ζ8ζ87ζ83ζ85ζ86ζ32ζ82ζ3ζ82ζ32ζ86ζ3ζ87ζ3ζ83ζ3ζ85ζ3ζ8ζ3ζ87ζ32ζ83ζ32ζ85ζ32ζ8ζ32    linear of order 24
ρ201-1ζ32ζ31ζ32ζ3-iiζ65ζ6ζ8ζ87ζ83ζ85ζ86ζ3ζ82ζ32ζ82ζ3ζ86ζ32ζ87ζ32ζ83ζ32ζ85ζ32ζ8ζ32ζ87ζ3ζ83ζ3ζ85ζ3ζ8ζ3    linear of order 24
ρ211-1ζ32ζ31ζ32ζ3i-iζ65ζ6ζ87ζ8ζ85ζ83ζ82ζ3ζ86ζ32ζ86ζ3ζ82ζ32ζ8ζ32ζ85ζ32ζ83ζ32ζ87ζ32ζ8ζ3ζ85ζ3ζ83ζ3ζ87ζ3    linear of order 24
ρ221-1ζ3ζ321ζ3ζ32i-iζ6ζ65ζ83ζ85ζ8ζ87ζ82ζ32ζ86ζ3ζ86ζ32ζ82ζ3ζ85ζ3ζ8ζ3ζ87ζ3ζ83ζ3ζ85ζ32ζ8ζ32ζ87ζ32ζ83ζ32    linear of order 24
ρ231-1ζ3ζ321ζ3ζ32i-iζ6ζ65ζ87ζ8ζ85ζ83ζ82ζ32ζ86ζ3ζ86ζ32ζ82ζ3ζ8ζ3ζ85ζ3ζ83ζ3ζ87ζ3ζ8ζ32ζ85ζ32ζ83ζ32ζ87ζ32    linear of order 24
ρ241-1ζ32ζ31ζ32ζ3i-iζ65ζ6ζ83ζ85ζ8ζ87ζ82ζ3ζ86ζ32ζ86ζ3ζ82ζ32ζ85ζ32ζ8ζ32ζ87ζ32ζ83ζ32ζ85ζ3ζ8ζ3ζ87ζ3ζ83ζ3    linear of order 24
ρ258088-1-1-100000000000000000000    orthogonal lifted from F9
ρ2680-4-4-3-4+4-3-1ζ6ζ6500000000000000000000    complex faithful
ρ2780-4+4-3-4-4-3-1ζ65ζ600000000000000000000    complex faithful

Permutation representations of C3×F9
On 24 points - transitive group 24T567
Generators in S24
(1 13 21)(2 14 22)(3 15 23)(4 16 24)(5 9 17)(6 10 18)(7 11 19)(8 12 20)
(2 14 22)(3 15 23)(4 24 16)(6 18 10)(7 19 11)(8 12 20)
(1 13 21)(3 15 23)(4 16 24)(5 17 9)(7 19 11)(8 20 12)
(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)| (1,13,21)(2,14,22)(3,15,23)(4,16,24)(5,9,17)(6,10,18)(7,11,19)(8,12,20), (2,14,22)(3,15,23)(4,24,16)(6,18,10)(7,19,11)(8,12,20), (1,13,21)(3,15,23)(4,16,24)(5,17,9)(7,19,11)(8,20,12), (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( (1,13,21)(2,14,22)(3,15,23)(4,16,24)(5,9,17)(6,10,18)(7,11,19)(8,12,20), (2,14,22)(3,15,23)(4,24,16)(6,18,10)(7,19,11)(8,12,20), (1,13,21)(3,15,23)(4,16,24)(5,17,9)(7,19,11)(8,20,12), (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([(1,13,21),(2,14,22),(3,15,23),(4,16,24),(5,9,17),(6,10,18),(7,11,19),(8,12,20)], [(2,14,22),(3,15,23),(4,24,16),(6,18,10),(7,19,11),(8,12,20)], [(1,13,21),(3,15,23),(4,16,24),(5,17,9),(7,19,11),(8,20,12)], [(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(24,567);

On 27 points - transitive group 27T78
Generators in S27
(1 3 2)(4 25 14)(5 26 15)(6 27 16)(7 20 17)(8 21 18)(9 22 19)(10 23 12)(11 24 13)
(1 11 7)(2 13 17)(3 24 20)(4 10 9)(5 6 8)(12 19 14)(15 16 18)(21 26 27)(22 25 23)
(1 4 8)(2 14 18)(3 25 21)(5 11 10)(6 7 9)(12 15 13)(16 17 19)(20 22 27)(23 26 24)
(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)| (1,3,2)(4,25,14)(5,26,15)(6,27,16)(7,20,17)(8,21,18)(9,22,19)(10,23,12)(11,24,13), (1,11,7)(2,13,17)(3,24,20)(4,10,9)(5,6,8)(12,19,14)(15,16,18)(21,26,27)(22,25,23), (1,4,8)(2,14,18)(3,25,21)(5,11,10)(6,7,9)(12,15,13)(16,17,19)(20,22,27)(23,26,24), (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( (1,3,2)(4,25,14)(5,26,15)(6,27,16)(7,20,17)(8,21,18)(9,22,19)(10,23,12)(11,24,13), (1,11,7)(2,13,17)(3,24,20)(4,10,9)(5,6,8)(12,19,14)(15,16,18)(21,26,27)(22,25,23), (1,4,8)(2,14,18)(3,25,21)(5,11,10)(6,7,9)(12,15,13)(16,17,19)(20,22,27)(23,26,24), (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([(1,3,2),(4,25,14),(5,26,15),(6,27,16),(7,20,17),(8,21,18),(9,22,19),(10,23,12),(11,24,13)], [(1,11,7),(2,13,17),(3,24,20),(4,10,9),(5,6,8),(12,19,14),(15,16,18),(21,26,27),(22,25,23)], [(1,4,8),(2,14,18),(3,25,21),(5,11,10),(6,7,9),(12,15,13),(16,17,19),(20,22,27),(23,26,24)], [(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(27,78);

C3×F9 is a maximal subgroup of   F9⋊S3

Matrix representation of C3×F9 in GL8(𝔽73)

640000000
064000000
006400000
000640000
000064000
000006400
000000640
000000064
,
10000000
01000000
10800000
900640000
6400064000
720000800
720000080
6400000064
,
640000000
658000000
650800000
000640000
80008000
000006400
90000010
90000001
,
720007000
000072100
000072010
000072001
00101000
00011000
01001000
00001000

G:=sub<GL(8,GF(73))| [64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64],[1,0,1,9,64,72,72,64,0,1,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,64],[64,65,65,0,8,0,9,9,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,7,72,72,72,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0] >;

C3×F9 in GAP, Magma, Sage, TeX

C_3\times F_9
% in TeX

G:=Group("C3xF9");
// GroupNames label

G:=SmallGroup(216,154);
// by ID

G=gap.SmallGroup(216,154);
# by ID

G:=PCGroup([6,-2,-3,-2,-2,-3,3,36,50,2164,856,142,6053,1169,455]);
// Polycyclic

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

Export

Subgroup lattice of C3×F9 in TeX
Character table of C3×F9 in TeX

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