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G = C3xS3xD13order 468 = 22·32·13

Direct product of C3, S3 and D13

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C3xS3xD13, C39:4D6, D39:3C6, C32:3D26, C13:4(S3xC6), C39:5(C2xC6), C3:1(C6xD13), (S3xC39):2C2, (S3xC13):3C6, (C3xD39):1C2, (C3xD13):3C6, (C3xC39):1C22, (C32xD13):1C2, SmallGroup(468,42)

Series: Derived Chief Lower central Upper central

C1C39 — C3xS3xD13
C1C13C39C3xC39C32xD13 — C3xS3xD13
C39 — C3xS3xD13
C1C3

Generators and relations for C3xS3xD13
 G = < a,b,c,d,e | a3=b3=c2=d13=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 336 in 44 conjugacy classes, 20 normal (all characteristic)
Quotients: C1, C2, C3, C22, S3, C6, D6, C2xC6, C3xS3, D13, S3xC6, D26, C3xD13, S3xD13, C6xD13, C3xS3xD13
3C2
13C2
39C2
2C3
39C22
3C6
13C6
13C6
13S3
26C6
39C6
3C26
3D13
2C39
13D6
39C2xC6
13C3xS3
13C3xC6
3D26
2C3xD13
3C3xD13
3C78
13S3xC6
3C6xD13

Smallest permutation representation of C3xS3xD13
On 78 points
Generators in S78
(1 21 35)(2 22 36)(3 23 37)(4 24 38)(5 25 39)(6 26 27)(7 14 28)(8 15 29)(9 16 30)(10 17 31)(11 18 32)(12 19 33)(13 20 34)(40 57 68)(41 58 69)(42 59 70)(43 60 71)(44 61 72)(45 62 73)(46 63 74)(47 64 75)(48 65 76)(49 53 77)(50 54 78)(51 55 66)(52 56 67)
(1 21 35)(2 22 36)(3 23 37)(4 24 38)(5 25 39)(6 26 27)(7 14 28)(8 15 29)(9 16 30)(10 17 31)(11 18 32)(12 19 33)(13 20 34)(40 68 57)(41 69 58)(42 70 59)(43 71 60)(44 72 61)(45 73 62)(46 74 63)(47 75 64)(48 76 65)(49 77 53)(50 78 54)(51 66 55)(52 67 56)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 40)(14 64)(15 65)(16 53)(17 54)(18 55)(19 56)(20 57)(21 58)(22 59)(23 60)(24 61)(25 62)(26 63)(27 74)(28 75)(29 76)(30 77)(31 78)(32 66)(33 67)(34 68)(35 69)(36 70)(37 71)(38 72)(39 73)
(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 55 56 57 58 59 60 61 62 63 64 65)(66 67 68 69 70 71 72 73 74 75 76 77 78)
(1 13)(2 12)(3 11)(4 10)(5 9)(6 8)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(27 29)(30 39)(31 38)(32 37)(33 36)(34 35)(40 41)(42 52)(43 51)(44 50)(45 49)(46 48)(53 62)(54 61)(55 60)(56 59)(57 58)(63 65)(66 71)(67 70)(68 69)(72 78)(73 77)(74 76)

G:=sub<Sym(78)| (1,21,35)(2,22,36)(3,23,37)(4,24,38)(5,25,39)(6,26,27)(7,14,28)(8,15,29)(9,16,30)(10,17,31)(11,18,32)(12,19,33)(13,20,34)(40,57,68)(41,58,69)(42,59,70)(43,60,71)(44,61,72)(45,62,73)(46,63,74)(47,64,75)(48,65,76)(49,53,77)(50,54,78)(51,55,66)(52,56,67), (1,21,35)(2,22,36)(3,23,37)(4,24,38)(5,25,39)(6,26,27)(7,14,28)(8,15,29)(9,16,30)(10,17,31)(11,18,32)(12,19,33)(13,20,34)(40,68,57)(41,69,58)(42,70,59)(43,71,60)(44,72,61)(45,73,62)(46,74,63)(47,75,64)(48,76,65)(49,77,53)(50,78,54)(51,66,55)(52,67,56), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,40)(14,64)(15,65)(16,53)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,74)(28,75)(29,76)(30,77)(31,78)(32,66)(33,67)(34,68)(35,69)(36,70)(37,71)(38,72)(39,73), (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)(53,54,55,56,57,58,59,60,61,62,63,64,65)(66,67,68,69,70,71,72,73,74,75,76,77,78), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(27,29)(30,39)(31,38)(32,37)(33,36)(34,35)(40,41)(42,52)(43,51)(44,50)(45,49)(46,48)(53,62)(54,61)(55,60)(56,59)(57,58)(63,65)(66,71)(67,70)(68,69)(72,78)(73,77)(74,76)>;

G:=Group( (1,21,35)(2,22,36)(3,23,37)(4,24,38)(5,25,39)(6,26,27)(7,14,28)(8,15,29)(9,16,30)(10,17,31)(11,18,32)(12,19,33)(13,20,34)(40,57,68)(41,58,69)(42,59,70)(43,60,71)(44,61,72)(45,62,73)(46,63,74)(47,64,75)(48,65,76)(49,53,77)(50,54,78)(51,55,66)(52,56,67), (1,21,35)(2,22,36)(3,23,37)(4,24,38)(5,25,39)(6,26,27)(7,14,28)(8,15,29)(9,16,30)(10,17,31)(11,18,32)(12,19,33)(13,20,34)(40,68,57)(41,69,58)(42,70,59)(43,71,60)(44,72,61)(45,73,62)(46,74,63)(47,75,64)(48,76,65)(49,77,53)(50,78,54)(51,66,55)(52,67,56), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,40)(14,64)(15,65)(16,53)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,74)(28,75)(29,76)(30,77)(31,78)(32,66)(33,67)(34,68)(35,69)(36,70)(37,71)(38,72)(39,73), (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)(53,54,55,56,57,58,59,60,61,62,63,64,65)(66,67,68,69,70,71,72,73,74,75,76,77,78), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(27,29)(30,39)(31,38)(32,37)(33,36)(34,35)(40,41)(42,52)(43,51)(44,50)(45,49)(46,48)(53,62)(54,61)(55,60)(56,59)(57,58)(63,65)(66,71)(67,70)(68,69)(72,78)(73,77)(74,76) );

G=PermutationGroup([[(1,21,35),(2,22,36),(3,23,37),(4,24,38),(5,25,39),(6,26,27),(7,14,28),(8,15,29),(9,16,30),(10,17,31),(11,18,32),(12,19,33),(13,20,34),(40,57,68),(41,58,69),(42,59,70),(43,60,71),(44,61,72),(45,62,73),(46,63,74),(47,64,75),(48,65,76),(49,53,77),(50,54,78),(51,55,66),(52,56,67)], [(1,21,35),(2,22,36),(3,23,37),(4,24,38),(5,25,39),(6,26,27),(7,14,28),(8,15,29),(9,16,30),(10,17,31),(11,18,32),(12,19,33),(13,20,34),(40,68,57),(41,69,58),(42,70,59),(43,71,60),(44,72,61),(45,73,62),(46,74,63),(47,75,64),(48,76,65),(49,77,53),(50,78,54),(51,66,55),(52,67,56)], [(1,41),(2,42),(3,43),(4,44),(5,45),(6,46),(7,47),(8,48),(9,49),(10,50),(11,51),(12,52),(13,40),(14,64),(15,65),(16,53),(17,54),(18,55),(19,56),(20,57),(21,58),(22,59),(23,60),(24,61),(25,62),(26,63),(27,74),(28,75),(29,76),(30,77),(31,78),(32,66),(33,67),(34,68),(35,69),(36,70),(37,71),(38,72),(39,73)], [(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),(53,54,55,56,57,58,59,60,61,62,63,64,65),(66,67,68,69,70,71,72,73,74,75,76,77,78)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(15,26),(16,25),(17,24),(18,23),(19,22),(20,21),(27,29),(30,39),(31,38),(32,37),(33,36),(34,35),(40,41),(42,52),(43,51),(44,50),(45,49),(46,48),(53,62),(54,61),(55,60),(56,59),(57,58),(63,65),(66,71),(67,70),(68,69),(72,78),(73,77),(74,76)]])

72 conjugacy classes

class 1 2A2B2C3A3B3C3D3E6A6B6C6D6E6F6G6H6I13A···13F26A···26F39A···39L39M···39AD78A···78L
order12223333366666666613···1326···2639···3939···3978···78
size1313391122233131326262639392···26···62···24···46···6

72 irreducible representations

dim111111112222222244
type+++++++++
imageC1C2C2C2C3C6C6C6S3D6C3xS3D13S3xC6D26C3xD13C6xD13S3xD13C3xS3xD13
kernelC3xS3xD13C32xD13S3xC39C3xD39S3xD13S3xC13C3xD13D39C3xD13C39D13C3xS3C13C32S3C3C3C1
# reps111122221126261212612

Matrix representation of C3xS3xD13 in GL4(F79) generated by

55000
05500
0010
0001
,
55000
02300
0010
0001
,
0100
1000
00780
00078
,
1000
0100
00481
00780
,
1000
0100
002567
005254
G:=sub<GL(4,GF(79))| [55,0,0,0,0,55,0,0,0,0,1,0,0,0,0,1],[55,0,0,0,0,23,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,78,0,0,0,0,78],[1,0,0,0,0,1,0,0,0,0,48,78,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,25,52,0,0,67,54] >;

C3xS3xD13 in GAP, Magma, Sage, TeX

C_3\times S_3\times D_{13}
% in TeX

G:=Group("C3xS3xD13");
// GroupNames label

G:=SmallGroup(468,42);
// by ID

G=gap.SmallGroup(468,42);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-13,248,10804]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^13=e^2=1,a*b=b*a,a*c=c*a,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=d^-1>;
// generators/relations

Export

Subgroup lattice of C3xS3xD13 in TeX

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