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G = C3⋊S3×3- 1+2order 486 = 2·35

Direct product of C3⋊S3 and 3- 1+2

direct product, metabelian, supersoluble, monomial

Aliases: C3⋊S3×3- 1+2, C34.7C6, (C32×C9)⋊26C6, C3⋊(S3×3- 1+2), C33.68(C3×S3), C33.55(C3×C6), C32.49(S3×C32), (C3×3- 1+2)⋊23S3, C326(C2×3- 1+2), (C32×3- 1+2)⋊5C2, C92(C3×C3⋊S3), (C9×C3⋊S3)⋊3C3, (C3×C9)⋊23(C3×S3), (C3×C3⋊S3).8C32, (C32×C3⋊S3).3C3, C32.21(C3×C3⋊S3), C3.10(C32×C3⋊S3), SmallGroup(486,233)

Series: Derived Chief Lower central Upper central

C1C33 — C3⋊S3×3- 1+2
C1C3C32C33C32×C9C32×3- 1+2 — C3⋊S3×3- 1+2
C32C33 — C3⋊S3×3- 1+2
C1C33- 1+2

Generators and relations for C3⋊S3×3- 1+2
 G = < a,b,c,d,e | a3=b3=c2=d9=e3=1, ab=ba, cac=a-1, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d4 >

Subgroups: 616 in 216 conjugacy classes, 49 normal (12 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, C32, C18, C3×S3, C3⋊S3, C3×C6, C3×C9, C3×C9, 3- 1+2, 3- 1+2, C33, C33, C33, S3×C9, C2×3- 1+2, S3×C32, C3×C3⋊S3, C3×C3⋊S3, C32×C9, C3×3- 1+2, C3×3- 1+2, C34, S3×3- 1+2, C9×C3⋊S3, C32×C3⋊S3, C32×3- 1+2, C3⋊S3×3- 1+2
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3⋊S3, C3×C6, 3- 1+2, C2×3- 1+2, S3×C32, C3×C3⋊S3, S3×3- 1+2, C32×C3⋊S3, C3⋊S3×3- 1+2

Smallest permutation representation of C3⋊S3×3- 1+2
On 54 points
Generators in S54
(1 38 28)(2 39 29)(3 40 30)(4 41 31)(5 42 32)(6 43 33)(7 44 34)(8 45 35)(9 37 36)(10 23 50)(11 24 51)(12 25 52)(13 26 53)(14 27 54)(15 19 46)(16 20 47)(17 21 48)(18 22 49)
(1 4 7)(2 5 8)(3 6 9)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 52 49)(47 53 50)(48 54 51)
(1 48)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 46)(9 47)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 28)(18 29)(19 45)(20 37)(21 38)(22 39)(23 40)(24 41)(25 42)(26 43)(27 44)
(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)(28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45)(46 47 48 49 50 51 52 53 54)
(2 8 5)(3 6 9)(10 13 16)(12 18 15)(19 25 22)(20 23 26)(29 35 32)(30 33 36)(37 40 43)(39 45 42)(46 52 49)(47 50 53)

G:=sub<Sym(54)| (1,38,28)(2,39,29)(3,40,30)(4,41,31)(5,42,32)(6,43,33)(7,44,34)(8,45,35)(9,37,36)(10,23,50)(11,24,51)(12,25,52)(13,26,53)(14,27,54)(15,19,46)(16,20,47)(17,21,48)(18,22,49), (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,52,49)(47,53,50)(48,54,51), (1,48)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,46)(9,47)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,28)(18,29)(19,45)(20,37)(21,38)(22,39)(23,40)(24,41)(25,42)(26,43)(27,44), (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)(28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54), (2,8,5)(3,6,9)(10,13,16)(12,18,15)(19,25,22)(20,23,26)(29,35,32)(30,33,36)(37,40,43)(39,45,42)(46,52,49)(47,50,53)>;

G:=Group( (1,38,28)(2,39,29)(3,40,30)(4,41,31)(5,42,32)(6,43,33)(7,44,34)(8,45,35)(9,37,36)(10,23,50)(11,24,51)(12,25,52)(13,26,53)(14,27,54)(15,19,46)(16,20,47)(17,21,48)(18,22,49), (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,31,34)(29,32,35)(30,33,36)(37,40,43)(38,41,44)(39,42,45)(46,52,49)(47,53,50)(48,54,51), (1,48)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,46)(9,47)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,28)(18,29)(19,45)(20,37)(21,38)(22,39)(23,40)(24,41)(25,42)(26,43)(27,44), (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)(28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54), (2,8,5)(3,6,9)(10,13,16)(12,18,15)(19,25,22)(20,23,26)(29,35,32)(30,33,36)(37,40,43)(39,45,42)(46,52,49)(47,50,53) );

G=PermutationGroup([[(1,38,28),(2,39,29),(3,40,30),(4,41,31),(5,42,32),(6,43,33),(7,44,34),(8,45,35),(9,37,36),(10,23,50),(11,24,51),(12,25,52),(13,26,53),(14,27,54),(15,19,46),(16,20,47),(17,21,48),(18,22,49)], [(1,4,7),(2,5,8),(3,6,9),(10,16,13),(11,17,14),(12,18,15),(19,25,22),(20,26,23),(21,27,24),(28,31,34),(29,32,35),(30,33,36),(37,40,43),(38,41,44),(39,42,45),(46,52,49),(47,53,50),(48,54,51)], [(1,48),(2,49),(3,50),(4,51),(5,52),(6,53),(7,54),(8,46),(9,47),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,28),(18,29),(19,45),(20,37),(21,38),(22,39),(23,40),(24,41),(25,42),(26,43),(27,44)], [(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),(28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45),(46,47,48,49,50,51,52,53,54)], [(2,8,5),(3,6,9),(10,13,16),(12,18,15),(19,25,22),(20,23,26),(29,35,32),(30,33,36),(37,40,43),(39,45,42),(46,52,49),(47,50,53)]])

66 conjugacy classes

class 1  2 3A3B3C···3N3O3P3Q···3X6A6B6C6D9A···9F9G···9AD18A···18F
order12333···3333···366669···99···918···18
size19112···2336···69927273···36···627···27

66 irreducible representations

dim111111222336
type+++
imageC1C2C3C3C6C6S3C3×S3C3×S33- 1+2C2×3- 1+2S3×3- 1+2
kernelC3⋊S3×3- 1+2C32×3- 1+2C9×C3⋊S3C32×C3⋊S3C32×C9C34C3×3- 1+2C3×C9C33C3⋊S3C32C3
# reps1162624248228

Matrix representation of C3⋊S3×3- 1+2 in GL7(𝔽19)

01800000
11800000
00181000
00180000
0000100
0000010
0000001
,
18100000
18000000
00181000
00180000
0000100
0000010
0000001
,
0100000
1000000
00118000
00018000
0000100
0000010
0000001
,
11000000
01100000
00110000
00011000
000018912
0000710
0000100
,
11000000
01100000
0070000
0007000
00001100
000018111
0000007

G:=sub<GL(7,GF(19))| [0,1,0,0,0,0,0,18,18,0,0,0,0,0,0,0,18,18,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[18,18,0,0,0,0,0,1,0,0,0,0,0,0,0,0,18,18,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,18,18,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[11,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,18,7,1,0,0,0,0,9,1,0,0,0,0,0,12,0,0],[11,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,11,18,0,0,0,0,0,0,1,0,0,0,0,0,0,11,7] >;

C3⋊S3×3- 1+2 in GAP, Magma, Sage, TeX

C_3\rtimes S_3\times 3_-^{1+2}
% in TeX

G:=Group("C3:S3xES-(3,1)");
// GroupNames label

G:=SmallGroup(486,233);
// by ID

G=gap.SmallGroup(486,233);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,224,68,3244,11669]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^9=e^3=1,a*b=b*a,c*a*c=a^-1,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^4>;
// generators/relations

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