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G = C345S3order 486 = 2·35

5th semidirect product of C34 and S3 acting faithfully

non-abelian, supersoluble, monomial

Aliases: C345S3, C3⋊C3≀S3, C3≀C33S3, He35(C3×S3), C335(C3⋊S3), (C3×He3)⋊10C6, He35S33C3, C33.43(C3×S3), C3.13(He34S3), C32.11(C32⋊C6), (C3×C3≀C3)⋊6C2, C32.2(C3×C3⋊S3), SmallGroup(486,166)

Series: Derived Chief Lower central Upper central

C1C3C3×He3 — C345S3
C1C3C32C33C3×He3C3×C3≀C3 — C345S3
C3×He3 — C345S3
C1C3

Generators and relations for C345S3
 G = < a,b,c,d,e,f | a3=b3=c3=d3=e3=f2=1, ab=ba, ac=ca, ad=da, eae-1=ac-1, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, ece-1=cd=dc, fcf=c-1d, de=ed, df=fd, fef=e-1 >

Subgroups: 956 in 144 conjugacy classes, 20 normal (12 characteristic)
C1, C2, C3 [×2], C3 [×13], S3 [×7], C6 [×4], C9 [×3], C32 [×2], C32 [×2], C32 [×33], C3×S3 [×19], C3⋊S3 [×2], C3×C6, C3×C9, He3 [×3], He3 [×2], 3- 1+2 [×5], C33 [×2], C33 [×11], He3⋊C2 [×5], S3×C32 [×4], C3×C3⋊S3 [×5], C3≀C3 [×3], C3≀C3 [×3], C3×He3, C3×3- 1+2, C34, C3≀S3 [×3], He35S3, C32×C3⋊S3, C3×C3≀C3, C345S3
Quotients: C1, C2, C3, S3 [×4], C6, C3×S3 [×4], C3⋊S3, C32⋊C6 [×3], C3×C3⋊S3, C3≀S3, He34S3, C345S3

Permutation representations of C345S3
On 18 points - transitive group 18T167
Generators in S18
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 2 3)(4 5 6)(7 9 8)(10 12 11)(13 15 14)(16 17 18)
(4 6 5)(7 9 8)(10 12 11)(16 18 17)
(1 3 2)(4 6 5)(7 8 9)(10 12 11)(13 15 14)(16 17 18)
(1 4 8)(2 5 7)(3 6 9)(10 17 13)(11 16 14)(12 18 15)
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)

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

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

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

G:=TransitiveGroup(18,167);

On 27 points - transitive group 27T194
Generators in S27
(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)
(1 9 5)(2 7 6)(3 8 4)(10 27 14)(11 25 15)(12 26 13)(16 20 23)(17 21 24)(18 19 22)
(1 7 4)(2 8 5)(3 9 6)(10 11 12)(13 14 15)(16 20 23)(17 21 24)(18 19 22)(25 26 27)
(1 3 2)(4 6 5)(7 9 8)(10 15 26)(11 13 27)(12 14 25)(16 21 22)(17 19 23)(18 20 24)
(1 22 13)(2 21 11)(3 16 27)(4 23 10)(5 19 26)(6 17 15)(7 24 25)(8 20 14)(9 18 12)
(4 8)(5 9)(6 7)(10 20)(11 21)(12 19)(13 22)(14 23)(15 24)(16 27)(17 25)(18 26)

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

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

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

G:=TransitiveGroup(27,194);

On 27 points - transitive group 27T195
Generators in S27
(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)
(1 7 4)(2 8 5)(3 9 6)(10 17 14)(11 18 15)(12 16 13)(19 27 23)(20 25 24)(21 26 22)
(1 2 3)(4 5 6)(7 8 9)(19 20 21)(22 23 24)(25 26 27)
(1 3 2)(4 6 5)(7 9 8)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)
(1 10 21)(2 12 20)(3 11 19)(4 14 22)(5 13 24)(6 15 23)(7 17 26)(8 16 25)(9 18 27)
(4 7)(5 8)(6 9)(10 21)(11 19)(12 20)(13 25)(14 26)(15 27)(16 24)(17 22)(18 23)

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

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

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

G:=TransitiveGroup(27,195);

39 conjugacy classes

class 1  2 3A3B3C3D3E3F···3K3L···3T3U3V3W6A···6H9A···9F
order12333333···33···33336···69···9
size127112223···36···618181827···2718···18

39 irreducible representations

dim11112222366
type+++++
imageC1C2C3C6S3S3C3×S3C3×S3C3≀S3C32⋊C6C345S3
kernelC345S3C3×C3≀C3He35S3C3×He3C3≀C3C34He3C33C3C32C1
# reps112231621236

Matrix representation of C345S3 in GL5(𝔽19)

10000
01000
00700
000110
000011
,
018000
118000
00100
00010
00001
,
10000
01000
001100
00070
00001
,
10000
01000
00700
00070
00007
,
10000
01000
00010
00001
00100
,
01000
10000
00100
00001
00010

G:=sub<GL(5,GF(19))| [1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,11,0,0,0,0,0,11],[0,1,0,0,0,18,18,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,11,0,0,0,0,0,7,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,7],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

C345S3 in GAP, Magma, Sage, TeX

C_3^4\rtimes_5S_3
% in TeX

G:=Group("C3^4:5S3");
// GroupNames label

G:=SmallGroup(486,166);
// by ID

G=gap.SmallGroup(486,166);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,218,548,867,8104,1096]);
// Polycyclic

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

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