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G = C343S3order 486 = 2·35

3rd semidirect product of C34 and S3 acting faithfully

non-abelian, supersoluble, monomial

Aliases: C343S3, (C3×He3)⋊9C6, (C3×He3)⋊7S3, C333(C3×S3), C32⋊He33C2, He35S32C3, C33.29(C3⋊S3), C322(C32⋊C6), C321(He3⋊C2), C3.11(He34S3), C32.35(C3×C3⋊S3), C3.2(C3×He3⋊C2), SmallGroup(486,145)

Series: Derived Chief Lower central Upper central

C1C3C3×He3 — C343S3
C1C3C32C33C3×He3C32⋊He3 — C343S3
C3×He3 — C343S3
C1C3

Generators and relations for C343S3
 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, ebe-1=bd=db, fbf=b-1, cd=dc, ce=ec, fcf=c-1, de=ed, df=fd, fef=e-1 >

Subgroups: 1172 in 192 conjugacy classes, 23 normal (11 characteristic)
C1, C2, C3 [×2], C3 [×16], S3 [×7], C6 [×4], C32, C32 [×6], C32 [×49], C3×S3 [×19], C3⋊S3 [×4], C3×C6, He3 [×24], C33 [×2], C33 [×3], C33 [×13], C32⋊C6 [×9], He3⋊C2 [×3], S3×C32 [×4], C3×C3⋊S3 [×7], C3×He3, C3×He3 [×3], C3×He3 [×4], C34, C3×C32⋊C6 [×3], He35S3, C32×C3⋊S3, C32⋊He3, C343S3
Quotients: C1, C2, C3, S3 [×4], C6, C3×S3 [×4], C3⋊S3, C32⋊C6 [×3], He3⋊C2 [×3], C3×C3⋊S3, He34S3, C3×He3⋊C2, C343S3

Permutation representations of C343S3
On 18 points - transitive group 18T165
Generators in S18
(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 2 3)(4 5 6)(13 15 14)(16 17 18)
(1 2 3)(4 5 6)(7 8 9)(10 12 11)(13 14 15)(16 18 17)
(1 3 2)(4 5 6)(7 8 9)(10 11 12)(13 15 14)(16 18 17)
(1 12 14)(2 11 15)(3 10 13)(4 17 7)(5 16 8)(6 18 9)
(1 4)(2 6)(3 5)(7 12)(8 10)(9 11)(13 16)(14 17)(15 18)

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

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

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

G:=TransitiveGroup(18,165);

On 18 points - transitive group 18T168
Generators in S18
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 15 12)(2 13 10)(3 14 11)(4 17 8)(5 18 9)(6 16 7)
(1 11 13)(2 12 14)(3 10 15)(4 9 16)(5 7 17)(6 8 18)
(1 11 13)(2 12 14)(3 10 15)(4 16 9)(5 17 7)(6 18 8)
(1 14 15)(2 3 11)(4 18 5)(6 7 9)(8 17 16)(10 13 12)
(1 6)(2 4)(3 5)(7 15)(8 13)(9 14)(10 17)(11 18)(12 16)

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

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

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

G:=TransitiveGroup(18,168);

On 27 points - transitive group 27T186
Generators in S27
(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)
(1 4 7)(2 5 8)(3 6 9)(10 27 13)(11 25 14)(12 26 15)(16 19 22)(17 20 23)(18 21 24)
(1 7 4)(2 8 5)(3 9 6)(10 11 12)(13 14 15)(16 18 17)(19 21 20)(22 24 23)(25 26 27)
(1 3 2)(4 6 5)(7 9 8)(10 15 25)(11 13 26)(12 14 27)(16 21 23)(17 19 24)(18 20 22)
(1 20 25)(2 18 15)(3 22 10)(4 21 27)(5 16 14)(6 23 12)(7 19 26)(8 17 13)(9 24 11)
(4 7)(5 8)(6 9)(10 22)(11 23)(12 24)(13 16)(14 17)(15 18)(19 27)(20 25)(21 26)

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

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

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

G:=TransitiveGroup(27,186);

39 conjugacy classes

class 1  2 3A3B3C3D3E3F···3K3L···3T3U···3AC6A···6H
order12333333···33···33···36···6
size127112223···36···618···1827···27

39 irreducible representations

dim1111222366
type+++++
imageC1C2C3C6S3S3C3×S3He3⋊C2C32⋊C6C343S3
kernelC343S3C32⋊He3He35S3C3×He3C3×He3C34C33C32C32C1
# reps11223181236

Matrix representation of C343S3 in GL6(𝔽7)

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

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

C343S3 in GAP, Magma, Sage, TeX

C_3^4\rtimes_3S_3
% in TeX

G:=Group("C3^4:3S3");
// GroupNames label

G:=SmallGroup(486,145);
// by ID

G=gap.SmallGroup(486,145);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,218,548,867,735,3244]);
// 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,e*b*e^-1=b*d=d*b,f*b*f=b^-1,c*d=d*c,c*e=e*c,f*c*f=c^-1,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations

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