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## G = C3×C32⋊C6order 162 = 2·34

### Direct product of C3 and C32⋊C6

Aliases: C3×C32⋊C6, He34C6, C332C6, C333S3, C3⋊S3⋊C32, C32⋊(C3×C6), (C3×He3)⋊1C2, C321(C3×S3), C3.2(S3×C32), (C3×C3⋊S3)⋊C3, SmallGroup(162,34)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3×C32⋊C6
 Chief series C1 — C3 — C32 — C33 — C3×He3 — C3×C32⋊C6
 Lower central C32 — C3×C32⋊C6
 Upper central C1 — C3

Generators and relations for C3×C32⋊C6
G = < a,b,c,d | a3=b3=c3=d6=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c-1, dcd-1=c-1 >

Subgroups: 200 in 58 conjugacy classes, 20 normal (11 characteristic)
C1, C2, C3 [×2], C3 [×9], S3 [×2], C6 [×4], C32 [×2], C32 [×3], C32 [×11], C3×S3 [×5], C3⋊S3, C3×C6, He3 [×3], He3 [×3], C33 [×2], C33, C32⋊C6 [×3], S3×C32, C3×C3⋊S3, C3×He3, C3×C32⋊C6
Quotients: C1, C2, C3 [×4], S3, C6 [×4], C32, C3×S3 [×4], C3×C6, C32⋊C6, S3×C32, C3×C32⋊C6

Character table of C3×C32⋊C6

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 3I 3J 3K 3L 3M 3N 3O 3P 3Q 3R 3S 3T 6A 6B 6C 6D 6E 6F 6G 6H size 1 9 1 1 2 2 2 3 3 3 3 3 3 6 6 6 6 6 6 6 6 6 9 9 9 9 9 9 9 9 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 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 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 ζ3 ζ32 ζ3 ζ32 1 ζ3 ζ3 ζ32 ζ32 1 1 ζ32 ζ3 1 ζ3 ζ32 ζ32 ζ3 1 1 ζ3 ζ32 ζ32 1 ζ3 ζ3 ζ32 1 linear of order 3 ρ4 1 -1 ζ3 ζ32 ζ3 ζ32 1 ζ32 1 ζ3 1 ζ3 ζ32 ζ32 1 1 ζ32 ζ3 1 ζ3 ζ3 ζ32 ζ65 ζ6 ζ65 ζ6 -1 ζ6 -1 ζ65 linear of order 6 ρ5 1 -1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 1 ζ32 1 ζ3 ζ32 ζ3 1 ζ3 ζ32 -1 -1 ζ6 ζ6 ζ6 ζ65 ζ65 ζ65 linear of order 6 ρ6 1 -1 ζ3 ζ32 ζ3 ζ32 1 ζ3 ζ3 ζ32 ζ32 1 1 ζ32 ζ3 1 ζ3 ζ32 ζ32 ζ3 1 1 ζ65 ζ6 ζ6 -1 ζ65 ζ65 ζ6 -1 linear of order 6 ρ7 1 1 ζ3 ζ32 ζ3 ζ32 1 ζ32 1 ζ3 1 ζ3 ζ32 ζ32 1 1 ζ32 ζ3 1 ζ3 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 1 ζ32 1 ζ3 linear of order 3 ρ8 1 1 ζ32 ζ3 ζ32 ζ3 1 ζ3 1 ζ32 1 ζ32 ζ3 ζ3 1 1 ζ3 ζ32 1 ζ32 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 1 ζ3 1 ζ32 linear of order 3 ρ9 1 -1 ζ32 ζ3 ζ32 ζ3 1 ζ32 ζ32 ζ3 ζ3 1 1 ζ3 ζ32 1 ζ32 ζ3 ζ3 ζ32 1 1 ζ6 ζ65 ζ65 -1 ζ6 ζ6 ζ65 -1 linear of order 6 ρ10 1 -1 ζ3 ζ32 ζ3 ζ32 1 1 ζ32 1 ζ3 ζ32 ζ3 ζ32 ζ32 1 1 1 ζ3 ζ3 ζ32 ζ3 ζ65 ζ6 -1 ζ65 ζ6 -1 ζ65 ζ6 linear of order 6 ρ11 1 1 ζ32 ζ3 ζ32 ζ3 1 ζ32 ζ32 ζ3 ζ3 1 1 ζ3 ζ32 1 ζ32 ζ3 ζ3 ζ32 1 1 ζ32 ζ3 ζ3 1 ζ32 ζ32 ζ3 1 linear of order 3 ρ12 1 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 1 ζ3 1 ζ32 ζ3 ζ32 1 ζ32 ζ3 1 1 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 linear of order 3 ρ13 1 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 1 ζ32 1 ζ3 ζ32 ζ3 1 ζ3 ζ32 1 1 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 linear of order 3 ρ14 1 1 ζ3 ζ32 ζ3 ζ32 1 1 ζ32 1 ζ3 ζ32 ζ3 ζ32 ζ32 1 1 1 ζ3 ζ3 ζ32 ζ3 ζ3 ζ32 1 ζ3 ζ32 1 ζ3 ζ32 linear of order 3 ρ15 1 -1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 1 ζ3 1 ζ32 ζ3 ζ32 1 ζ32 ζ3 -1 -1 ζ65 ζ65 ζ65 ζ6 ζ6 ζ6 linear of order 6 ρ16 1 -1 ζ32 ζ3 ζ32 ζ3 1 ζ3 1 ζ32 1 ζ32 ζ3 ζ3 1 1 ζ3 ζ32 1 ζ32 ζ32 ζ3 ζ6 ζ65 ζ6 ζ65 -1 ζ65 -1 ζ6 linear of order 6 ρ17 1 1 ζ32 ζ3 ζ32 ζ3 1 1 ζ3 1 ζ32 ζ3 ζ32 ζ3 ζ3 1 1 1 ζ32 ζ32 ζ3 ζ32 ζ32 ζ3 1 ζ32 ζ3 1 ζ32 ζ3 linear of order 3 ρ18 1 -1 ζ32 ζ3 ζ32 ζ3 1 1 ζ3 1 ζ32 ζ3 ζ32 ζ3 ζ3 1 1 1 ζ32 ζ32 ζ3 ζ32 ζ6 ζ65 -1 ζ6 ζ65 -1 ζ6 ζ65 linear of order 6 ρ19 2 0 2 2 2 2 2 2 2 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 0 0 orthogonal lifted from S3 ρ20 2 0 -1-√-3 -1+√-3 -1-√-3 -1+√-3 2 2 -1+√-3 2 -1-√-3 -1+√-3 -1-√-3 ζ65 ζ65 -1 -1 -1 ζ6 ζ6 ζ65 ζ6 0 0 0 0 0 0 0 0 complex lifted from C3×S3 ρ21 2 0 -1-√-3 -1+√-3 -1-√-3 -1+√-3 2 -1-√-3 -1-√-3 -1+√-3 -1+√-3 2 2 ζ65 ζ6 -1 ζ6 ζ65 ζ65 ζ6 -1 -1 0 0 0 0 0 0 0 0 complex lifted from C3×S3 ρ22 2 0 2 2 2 2 2 -1+√-3 -1-√-3 -1-√-3 -1+√-3 -1+√-3 -1-√-3 -1 ζ6 -1 ζ65 ζ6 ζ65 -1 ζ65 ζ6 0 0 0 0 0 0 0 0 complex lifted from C3×S3 ρ23 2 0 2 2 2 2 2 -1-√-3 -1+√-3 -1+√-3 -1-√-3 -1-√-3 -1+√-3 -1 ζ65 -1 ζ6 ζ65 ζ6 -1 ζ6 ζ65 0 0 0 0 0 0 0 0 complex lifted from C3×S3 ρ24 2 0 -1+√-3 -1-√-3 -1+√-3 -1-√-3 2 -1+√-3 -1+√-3 -1-√-3 -1-√-3 2 2 ζ6 ζ65 -1 ζ65 ζ6 ζ6 ζ65 -1 -1 0 0 0 0 0 0 0 0 complex lifted from C3×S3 ρ25 2 0 -1+√-3 -1-√-3 -1+√-3 -1-√-3 2 -1-√-3 2 -1+√-3 2 -1+√-3 -1-√-3 ζ6 -1 -1 ζ6 ζ65 -1 ζ65 ζ65 ζ6 0 0 0 0 0 0 0 0 complex lifted from C3×S3 ρ26 2 0 -1+√-3 -1-√-3 -1+√-3 -1-√-3 2 2 -1-√-3 2 -1+√-3 -1-√-3 -1+√-3 ζ6 ζ6 -1 -1 -1 ζ65 ζ65 ζ6 ζ65 0 0 0 0 0 0 0 0 complex lifted from C3×S3 ρ27 2 0 -1-√-3 -1+√-3 -1-√-3 -1+√-3 2 -1+√-3 2 -1-√-3 2 -1-√-3 -1+√-3 ζ65 -1 -1 ζ65 ζ6 -1 ζ6 ζ6 ζ65 0 0 0 0 0 0 0 0 complex lifted from C3×S3 ρ28 6 0 6 6 -3 -3 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊C6 ρ29 6 0 -3+3√-3 -3-3√-3 3-3√-3/2 3+3√-3/2 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ30 6 0 -3-3√-3 -3+3√-3 3+3√-3/2 3-3√-3/2 -3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of C3×C32⋊C6
On 18 points - transitive group 18T76
Generators in S18
(1 8 17)(2 9 18)(3 10 13)(4 11 14)(5 12 15)(6 7 16)
(1 8 17)(3 13 10)(4 14 11)(6 7 16)
(1 17 8)(2 9 18)(3 13 10)(4 11 14)(5 15 12)(6 7 16)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)

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

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

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

G:=TransitiveGroup(18,76);

On 18 points - transitive group 18T78
Generators in S18
(1 4 6)(2 3 5)(7 9 11)(8 10 12)(13 17 15)(14 18 16)
(1 8 16)(2 13 11)(3 17 7)(4 10 14)(5 15 9)(6 12 18)
(1 4 6)(2 5 3)(7 11 9)(8 10 12)(13 15 17)(14 18 16)
(1 2)(3 4)(5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)

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

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

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

G:=TransitiveGroup(18,78);

On 18 points - transitive group 18T81
Generators in S18
(1 14 7)(2 15 8)(3 16 9)(4 17 10)(5 18 11)(6 13 12)
(1 11 3)(2 17 13)(4 6 8)(5 16 14)(7 18 9)(10 12 15)
(1 7 14)(2 15 8)(3 9 16)(4 17 10)(5 11 18)(6 13 12)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)

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

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

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

G:=TransitiveGroup(18,81);

On 27 points - transitive group 27T48
Generators in S27
(1 7 5)(2 8 6)(3 9 4)(10 20 27)(11 21 22)(12 16 23)(13 17 24)(14 18 25)(15 19 26)
(1 10 13)(3 15 12)(4 26 23)(5 27 24)(7 20 17)(9 19 16)
(1 13 10)(2 11 14)(3 15 12)(4 26 23)(5 24 27)(6 22 25)(7 17 20)(8 21 18)(9 19 16)
(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)

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

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

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

G:=TransitiveGroup(27,48);

On 27 points - transitive group 27T60
Generators in S27
(1 3 2)(4 15 22)(5 10 23)(6 11 24)(7 12 25)(8 13 26)(9 14 27)(16 18 20)(17 19 21)
(1 13 10)(2 8 5)(3 26 23)(4 27 20)(6 12 19)(7 17 24)(9 16 15)(11 25 21)(14 18 22)
(1 20 17)(2 18 21)(3 16 19)(4 24 13)(5 14 25)(6 26 15)(7 10 27)(8 22 11)(9 12 23)
(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)

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

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

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

G:=TransitiveGroup(27,60);

C3×C32⋊C6 is a maximal subgroup of
He35D6  C3.C3≀S3  C32⋊C9⋊C6  C3.3C3≀S3  C34⋊C6  C9⋊He3⋊C2  (C3×He3)⋊C6  C9⋊S3⋊C32  He3.(C3×S3)  C343S3  (C32×C9)⋊S3  C33⋊(C3×S3)  He3.C32C6  He3⋊(C3×S3)  3+ 1+4⋊C2  3- 1+4⋊C2
C3×C32⋊C6 is a maximal quotient of
C34⋊C6  C34⋊S3  C34.C6  C34.S3  C9⋊He3⋊C2  C3≀S33C3  C3≀C3⋊C6  (C3×He3)⋊C6  He3.C3⋊C6  C9⋊S3⋊C32  He3.(C3×C6)  He3.(C3×S3)  C3≀C3.C6

Matrix representation of C3×C32⋊C6 in GL6(𝔽7)

 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 4 0 0 0 0 0 0 2 0 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 4 0 0 0 0 0 0 1
,
 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 2 0 0 0 2 0 0 0 0 0 0 2 0 0 0 2 0 0 0 0 0

G:=sub<GL(6,GF(7))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,0,0,0,0,2,0,0,0,2,0,0,0,0,0,0,2,0,0,0,2,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0] >;

C3×C32⋊C6 in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes C_6
% in TeX

G:=Group("C3xC3^2:C6");
// GroupNames label

G:=SmallGroup(162,34);
// by ID

G=gap.SmallGroup(162,34);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,723,728,2704]);
// Polycyclic

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

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