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G = C331D9order 486 = 2·35

1st semidirect product of C33 and D9 acting via D9/C3=S3

non-abelian, supersoluble, monomial

Aliases: C331D9, C34.1S3, C3.4C3≀S3, C32⋊C93C6, C33⋊C92C2, C32.5(C3×D9), C322D94C3, C32.2(C9⋊C6), C33.26(C3×S3), C3.6(C32⋊D9), C32.35(C32⋊C6), SmallGroup(486,19)

Series: Derived Chief Lower central Upper central

C1C3C32⋊C9 — C331D9
C1C3C32C33C32⋊C9C33⋊C9 — C331D9
C32⋊C9 — C331D9
C1C3

Generators and relations for C331D9
 G = < a,b,c,d,e | a3=b3=c3=d9=e2=1, ab=ba, ac=ca, dad-1=eae=abc-1, bc=cb, dbd-1=bc-1, ebe=b-1c-1, cd=dc, ce=ec, ede=d-1 >

Subgroups: 668 in 105 conjugacy classes, 14 normal (12 characteristic)
C1, C2, C3 [×2], C3 [×10], S3 [×4], C6 [×4], C9 [×2], C32 [×2], C32 [×2], C32 [×28], D9, C3×S3 [×16], C3⋊S3, C3×C6, C3×C9 [×2], C33 [×2], C33 [×10], C3×D9, S3×C32 [×4], C3×C3⋊S3 [×4], C32⋊C9, C32⋊C9, C34, C322D9, C32×C3⋊S3, C33⋊C9, C331D9
Quotients: C1, C2, C3, S3, C6, D9, C3×S3, C3×D9, C32⋊C6, C9⋊C6 [×2], C32⋊D9, C3≀S3, C331D9

Permutation representations of C331D9
On 18 points - transitive group 18T172
Generators in S18
(2 8 5)(12 15 18)
(2 8 5)(3 6 9)(10 16 13)(12 15 18)
(1 4 7)(2 5 8)(3 6 9)(10 16 13)(11 17 14)(12 18 15)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)
(1 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)(9 10)

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

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

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

G:=TransitiveGroup(18,172);

On 27 points - transitive group 27T188
Generators in S27
(2 22 18)(3 26 16)(5 25 12)(6 20 10)(8 19 15)(9 23 13)
(1 14 24)(2 22 18)(3 6 9)(4 17 27)(5 25 12)(7 11 21)(8 19 15)(10 13 16)(20 23 26)
(1 27 11)(2 19 12)(3 20 13)(4 21 14)(5 22 15)(6 23 16)(7 24 17)(8 25 18)(9 26 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)
(1 9)(2 8)(3 7)(4 6)(10 11)(12 18)(13 17)(14 16)(19 25)(20 24)(21 23)(26 27)

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

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

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

G:=TransitiveGroup(27,188);

On 27 points - transitive group 27T189
Generators in S27
(2 19 12)(3 20 13)(5 22 15)(6 23 16)(8 25 18)(9 26 10)
(1 11 27)(2 19 12)(4 14 21)(5 22 15)(7 17 24)(8 25 18)
(1 27 11)(2 19 12)(3 20 13)(4 21 14)(5 22 15)(6 23 16)(7 24 17)(8 25 18)(9 26 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)
(1 9)(2 8)(3 7)(4 6)(10 11)(12 18)(13 17)(14 16)(19 25)(20 24)(21 23)(26 27)

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

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

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

G:=TransitiveGroup(27,189);

39 conjugacy classes

class 1  2 3A3B3C3D3E3F···3K3L···3T6A···6H9A···9I
order12333333···33···36···69···9
size127112223···36···627···2718···18

39 irreducible representations

dim111122223666
type++++++
imageC1C2C3C6S3D9C3×S3C3×D9C3≀S3C32⋊C6C9⋊C6C331D9
kernelC331D9C33⋊C9C322D9C32⋊C9C34C33C33C32C3C32C32C1
# reps1122132612126

Matrix representation of C331D9 in GL5(𝔽19)

10000
01000
001100
001270
001207
,
10000
01000
00700
007110
001801
,
10000
01000
001100
000110
000011
,
111000
1215000
00106
000018
000118
,
012000
80000
00106
000118
000018

G:=sub<GL(5,GF(19))| [1,0,0,0,0,0,1,0,0,0,0,0,11,12,12,0,0,0,7,0,0,0,0,0,7],[1,0,0,0,0,0,1,0,0,0,0,0,7,7,18,0,0,0,11,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,11,0,0,0,0,0,11,0,0,0,0,0,11],[11,12,0,0,0,1,15,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,6,18,18],[0,8,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,6,18,18] >;

C331D9 in GAP, Magma, Sage, TeX

C_3^3\rtimes_1D_9
% in TeX

G:=Group("C3^3:1D9");
// GroupNames label

G:=SmallGroup(486,19);
// by ID

G=gap.SmallGroup(486,19);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1190,224,338,4755,735,3244]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^9=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e=a*b*c^-1,b*c=c*b,d*b*d^-1=b*c^-1,e*b*e=b^-1*c^-1,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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