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## G = C3×C32⋊D6order 324 = 22·34

### Direct product of C3 and C32⋊D6

Aliases: C3×C32⋊D6, C333D6, C32⋊C6⋊C6, C32⋊(S3×C6), He32(C2×C6), C32.12S32, He3⋊C23C6, (C3×He3)⋊2C22, C3⋊S3⋊(C3×S3), C3.4(C3×S32), (C3×C3⋊S3)⋊1S3, (C3×C32⋊C6)⋊2C2, (C3×He3⋊C2)⋊1C2, SmallGroup(324,117)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — C3×C32⋊D6
 Chief series C1 — C3 — C32 — He3 — C3×He3 — C3×C32⋊C6 — C3×C32⋊D6
 Lower central He3 — C3×C32⋊D6
 Upper central C1 — C3

Generators and relations for C3×C32⋊D6
G = < a,b,c,d,e | a3=b3=c3=d6=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe=b-1c-1, dcd-1=c-1, ce=ec, ede=d-1 >

Subgroups: 546 in 103 conjugacy classes, 22 normal (10 characteristic)
C1, C2 [×3], C3 [×2], C3 [×7], C22, S3 [×7], C6 [×9], C32, C32 [×2], C32 [×11], D6 [×3], C2×C6, C3×S3 [×18], C3⋊S3 [×2], C3×C6 [×3], He3, He3 [×3], C33 [×2], C33, S32 [×2], S3×C6 [×3], C32⋊C6 [×2], C32⋊C6 [×2], He3⋊C2, S3×C32 [×5], C3×C3⋊S3 [×2], C3×He3, C32⋊D6, C3×S32 [×2], C3×C32⋊C6 [×2], C3×He3⋊C2, C3×C32⋊D6
Quotients: C1, C2 [×3], C3, C22, S3 [×2], C6 [×3], D6 [×2], C2×C6, C3×S3 [×2], S32, S3×C6 [×2], C32⋊D6, C3×S32, C3×C32⋊D6

Permutation representations of C3×C32⋊D6
On 18 points - transitive group 18T118
Generators in S18
(1 17 11)(2 18 12)(3 13 7)(4 14 8)(5 15 9)(6 16 10)
(1 9 7)(2 14 6)(3 17 5)(4 10 12)(8 16 18)(11 15 13)
(1 17 11)(2 12 18)(3 13 7)(4 8 14)(5 15 9)(6 10 16)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
(1 3)(4 6)(7 11)(8 10)(13 17)(14 16)

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

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

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

G:=TransitiveGroup(18,118);

On 18 points - transitive group 18T126
Generators in S18
(1 12 14)(2 7 15)(3 8 16)(4 9 17)(5 10 18)(6 11 13)
(2 7 15)(3 8 16)(5 18 10)(6 13 11)
(1 14 12)(2 7 15)(3 16 8)(4 9 17)(5 18 10)(6 11 13)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
(1 3)(4 6)(8 12)(9 11)(13 17)(14 16)

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

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

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

G:=TransitiveGroup(18,126);

On 27 points - transitive group 27T125
Generators in S27
(1 4 9)(2 5 7)(3 6 8)(10 19 24)(11 20 25)(12 21 26)(13 16 27)(14 17 22)(15 18 23)
(2 26 23)(3 27 24)(5 12 15)(6 13 10)(7 21 18)(8 16 19)
(1 22 25)(2 26 23)(3 24 27)(4 14 11)(5 12 15)(6 10 13)(7 21 18)(8 19 16)(9 17 20)
(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)
(1 3)(4 6)(8 9)(10 14)(11 13)(16 20)(17 19)(22 24)(25 27)

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

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

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

G:=TransitiveGroup(27,125);

33 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F ··· 3K 3L 3M 3N 6A ··· 6F 6G ··· 6O order 1 2 2 2 3 3 3 3 3 3 ··· 3 3 3 3 6 ··· 6 6 ··· 6 size 1 9 9 9 1 1 2 2 2 6 ··· 6 12 12 12 9 ··· 9 18 ··· 18

33 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 6 6 type + + + + + + + image C1 C2 C2 C3 C6 C6 S3 D6 C3×S3 S3×C6 S32 C3×S32 C32⋊D6 C3×C32⋊D6 kernel C3×C32⋊D6 C3×C32⋊C6 C3×He3⋊C2 C32⋊D6 C32⋊C6 He3⋊C2 C3×C3⋊S3 C33 C3⋊S3 C32 C32 C3 C3 C1 # reps 1 2 1 2 4 2 2 2 4 4 1 2 2 4

Matrix representation of C3×C32⋊D6 in GL6(𝔽7)

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

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

C3×C32⋊D6 in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes D_6
% in TeX

G:=Group("C3xC3^2:D6");
// GroupNames label

G:=SmallGroup(324,117);
// by ID

G=gap.SmallGroup(324,117);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,297,2164,382,7781,3899]);
// Polycyclic

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

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