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G = C34.C6order 486 = 2·35

4th non-split extension by C34 of C6 acting faithfully

metabelian, supersoluble, monomial

Aliases: C34.4C6, C32⋊C97C6, C32⋊C181C3, C33.23(C3×C6), C33.55(C3×S3), C34.C33C2, C3⋊S323- 1+2, C32.37(S3×C32), C3.2(S3×3- 1+2), C32.20(C32⋊C6), (C3×3- 1+2)⋊13S3, C322(C2×3- 1+2), (C3×C9)⋊5(C3×S3), (C32×C3⋊S3).2C3, (C3×C3⋊S3).4C32, C3.13(C3×C32⋊C6), SmallGroup(486,104)

Series: Derived Chief Lower central Upper central

C1C33 — C34.C6
C1C3C32C33C34C34.C3 — C34.C6
C32C33 — C34.C6
C1C3C32

Generators and relations for C34.C6
 G = < a,b,c,d,e | a3=b3=c3=d3=1, e6=b, 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: 472 in 108 conjugacy classes, 24 normal (15 characteristic)
C1, C2, C3 [×2], C3 [×8], S3 [×2], C6 [×2], C9 [×6], C32 [×3], C32 [×27], C18 [×3], C3×S3 [×6], C3⋊S3, C3×C6, C3×C9 [×3], C3×C9 [×3], 3- 1+2 [×5], C33 [×2], C33 [×8], S3×C9 [×3], C2×3- 1+2, S3×C32 [×2], C3×C3⋊S3, C3×C3⋊S3, C32⋊C9 [×3], C32⋊C9 [×3], C3×3- 1+2, C3×3- 1+2, C34, C32⋊C18 [×3], S3×3- 1+2, C32×C3⋊S3, C34.C3, C34.C6
Quotients: C1, C2, C3 [×4], S3, C6 [×4], C32, C3×S3 [×4], C3×C6, 3- 1+2, C32⋊C6, C2×3- 1+2, S3×C32, C3×C32⋊C6, S3×3- 1+2, C34.C6

Permutation representations of C34.C6
On 18 points - transitive group 18T160
Generators in S18
(2 8 14)(3 15 9)(5 11 17)(6 18 12)
(1 7 13)(2 8 14)(3 9 15)(4 10 16)(5 11 17)(6 12 18)
(2 8 14)(3 9 15)(5 17 11)(6 18 12)
(1 13 7)(2 8 14)(3 15 9)(4 10 16)(5 17 11)(6 12 18)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18)

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

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

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

G:=TransitiveGroup(18,160);

On 27 points - transitive group 27T208
Generators in S27
(2 8 5)(3 6 9)(10 16 22)(11 23 17)(13 19 25)(14 26 20)
(1 7 4)(2 8 5)(3 9 6)(10 16 22)(11 17 23)(12 18 24)(13 19 25)(14 20 26)(15 21 27)
(2 10 19)(3 11 20)(5 22 13)(6 23 14)(8 16 25)(9 17 26)
(1 18 27)(2 10 19)(3 20 11)(4 12 21)(5 22 13)(6 14 23)(7 24 15)(8 16 25)(9 26 17)
(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)| (2,8,5)(3,6,9)(10,16,22)(11,23,17)(13,19,25)(14,26,20), (1,7,4)(2,8,5)(3,9,6)(10,16,22)(11,17,23)(12,18,24)(13,19,25)(14,20,26)(15,21,27), (2,10,19)(3,11,20)(5,22,13)(6,23,14)(8,16,25)(9,17,26), (1,18,27)(2,10,19)(3,20,11)(4,12,21)(5,22,13)(6,14,23)(7,24,15)(8,16,25)(9,26,17), (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( (2,8,5)(3,6,9)(10,16,22)(11,23,17)(13,19,25)(14,26,20), (1,7,4)(2,8,5)(3,9,6)(10,16,22)(11,17,23)(12,18,24)(13,19,25)(14,20,26)(15,21,27), (2,10,19)(3,11,20)(5,22,13)(6,23,14)(8,16,25)(9,17,26), (1,18,27)(2,10,19)(3,20,11)(4,12,21)(5,22,13)(6,14,23)(7,24,15)(8,16,25)(9,26,17), (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([(2,8,5),(3,6,9),(10,16,22),(11,23,17),(13,19,25),(14,26,20)], [(1,7,4),(2,8,5),(3,9,6),(10,16,22),(11,17,23),(12,18,24),(13,19,25),(14,20,26),(15,21,27)], [(2,10,19),(3,11,20),(5,22,13),(6,23,14),(8,16,25),(9,17,26)], [(1,18,27),(2,10,19),(3,20,11),(4,12,21),(5,22,13),(6,14,23),(7,24,15),(8,16,25),(9,26,17)], [(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(27,208);

42 conjugacy classes

class 1  2 3A3B3C3D3E3F3G3H···3R6A6B6C6D9A···9F9G···9L18A···18F
order1233333333···366669···99···918···18
size1911222336···69927279···918···1827···27

42 irreducible representations

dim111111222336666
type++++
imageC1C2C3C3C6C6S3C3×S3C3×S33- 1+2C2×3- 1+2C32⋊C6C3×C32⋊C6S3×3- 1+2C34.C6
kernelC34.C6C34.C3C32⋊C18C32×C3⋊S3C32⋊C9C34C3×3- 1+2C3×C9C33C3⋊S3C32C32C3C3C1
# reps116262162221226

Matrix representation of C34.C6 in GL6(𝔽19)

1100000
010000
1817000
0601100
400010
14140007
,
700000
070000
007000
000700
000070
000007
,
100000
0110000
11127000
0130100
9170070
3000011
,
1100000
0110000
0011000
290700
13170070
14160007
,
141600015
14601000
2501111
8171113017
121101406
1060205

G:=sub<GL(6,GF(19))| [11,0,18,0,4,14,0,1,1,6,0,14,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,7],[7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[1,0,11,0,9,3,0,11,12,13,17,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,11],[11,0,0,2,13,14,0,11,0,9,17,16,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[14,14,2,8,12,10,16,6,5,17,11,6,0,0,0,11,0,0,0,10,11,13,14,2,0,0,1,0,0,0,15,0,1,17,6,5] >;

C34.C6 in GAP, Magma, Sage, TeX

C_3^4.C_6
% in TeX

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

G:=SmallGroup(486,104);
// by ID

G=gap.SmallGroup(486,104);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,115,224,3244,3250,11669]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^3=1,e^6=b,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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