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G = C34⋊C6order 486 = 2·35

1st semidirect product of C34 and C6 acting faithfully

metabelian, supersoluble, monomial

Aliases: C341C6, C3⋊S3⋊He3, C32⋊(C2×He3), (C3×He3)⋊2C6, (C3×He3)⋊3S3, C331(C3×S3), C32⋊He31C2, C3.2(S3×He3), C33.22(C3×C6), C324(C32⋊C6), C32.35(S3×C32), (C3×C32⋊C6)⋊C3, (C32×C3⋊S3)⋊1C3, (C3×C3⋊S3).3C32, C3.12(C3×C32⋊C6), SmallGroup(486,102)

Series: Derived Chief Lower central Upper central

C1C33 — C34⋊C6
C1C3C32C33C34C32⋊He3 — C34⋊C6
C32C33 — C34⋊C6
C1C3C32

Generators and relations for C34⋊C6
 G = < a,b,c,d,e | a3=b3=c3=d3=e6=1, eae-1=ab=ba, ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1d-1, ede-1=d-1 >

Subgroups: 832 in 153 conjugacy classes, 24 normal (15 characteristic)
C1, C2, C3 [×2], C3 [×14], S3 [×2], C6 [×5], C32 [×3], C32 [×48], C3×S3 [×9], C3⋊S3, C3×C6 [×4], He3 [×20], C33 [×2], C33 [×3], C33 [×11], C32⋊C6 [×3], C2×He3, S3×C32 [×5], C3×C3⋊S3, C3×C3⋊S3, C3×He3, C3×He3 [×3], C3×He3 [×4], C34, C3×C32⋊C6 [×3], S3×He3, C32×C3⋊S3, C32⋊He3, C34⋊C6
Quotients: C1, C2, C3 [×4], S3, C6 [×4], C32, C3×S3 [×4], C3×C6, He3, C32⋊C6, C2×He3, S3×C32, C3×C32⋊C6, S3×He3, C34⋊C6

Permutation representations of C34⋊C6
On 18 points - transitive group 18T161
Generators in S18
(1 14 9)(2 17 12)(3 18 11)(4 15 8)(5 16 7)(6 13 10)
(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 15 17)(14 16 18)
(1 14 9)(2 12 17)(3 18 11)(4 8 15)(5 16 7)(6 10 13)
(1 5 3)(2 4 6)(7 11 9)(8 10 12)(13 17 15)(14 16 18)
(1 2)(3 4)(5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)

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

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

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

G:=TransitiveGroup(18,161);

On 18 points - transitive group 18T164
Generators in S18
(1 8 14)(2 15 9)(4 11 17)(5 18 12)
(1 8 14)(2 9 15)(3 10 16)(4 11 17)(5 12 18)(6 7 13)
(2 9 15)(3 10 16)(5 18 12)(6 13 7)
(1 14 8)(2 9 15)(3 16 10)(4 11 17)(5 18 12)(6 7 13)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)

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

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

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

G:=TransitiveGroup(18,164);

On 27 points - transitive group 27T180
Generators in S27
(1 5 9)(2 7 6)(10 19 26)(11 27 20)(13 16 23)(14 24 17)
(1 5 9)(2 6 7)(3 4 8)(10 19 26)(11 20 27)(12 21 22)(13 16 23)(14 17 24)(15 18 25)
(1 16 19)(2 17 20)(5 23 26)(6 24 27)(7 14 11)(9 13 10)
(1 16 19)(2 20 17)(3 18 21)(4 25 22)(5 23 26)(6 27 24)(7 11 14)(8 15 12)(9 13 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 25 26 27)

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

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

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

G:=TransitiveGroup(27,180);

42 conjugacy classes

class 1  2 3A3B3C3D3E3F3G3H···3R3S···3X3Y···3AD6A6B6C···6J
order1233333333···33···33···3666···6
size1911222336···69···918···189927···27

42 irreducible representations

dim11111122336666
type++++
imageC1C2C3C3C6C6S3C3×S3He3C2×He3C32⋊C6C3×C32⋊C6S3×He3C34⋊C6
kernelC34⋊C6C32⋊He3C3×C32⋊C6C32×C3⋊S3C3×He3C34C3×He3C33C3⋊S3C32C32C3C3C1
# reps11626218221226

Matrix representation of C34⋊C6 in GL6(𝔽7)

001000
100000
010000
000001
000100
000010
,
400000
040000
004000
000400
000040
000004
,
010000
001000
100000
000001
000100
000010
,
200000
020000
002000
000400
000040
000004
,
000100
000020
000004
100000
020000
004000

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

C34⋊C6 in GAP, Magma, Sage, TeX

C_3^4\rtimes C_6
% in TeX

G:=Group("C3^4:C6");
// GroupNames label

G:=SmallGroup(486,102);
// by ID

G=gap.SmallGroup(486,102);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,224,3244,3250,11669]);
// Polycyclic

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

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