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G = C3×S3×D13  order 468 = 22·32·13

Direct product of C3, S3 and D13

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

Aliases: C3×S3×D13, C39⋊4D6, D39⋊3C6, C32⋊3D26, C13⋊4(S3×C6), C39⋊5(C2×C6), C3⋊1(C6×D13), (S3×C39)⋊2C2, (S3×C13)⋊3C6, (C3×D39)⋊1C2, (C3×D13)⋊3C6, (C3×C39)⋊1C22, (C32×D13)⋊1C2, SmallGroup(468,42)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C39 — C3×S3×D13
Chief series
C1 — C13 — C39 — C3×C39 — C32×D13 — C3×S3×D13
Lower central
C39 — C3×S3×D13
Upper central
C1 — C3

Generators and relations for C3×S3×D13
 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 >

C1
3C2
13C2
39C2
C3
C3
2C3
C13
39C22
S3
3C6
13C6
13C6
13S3
26C6
39C6
C32
D13
3C26
3D13
C39
C39
2C39
13D6
39C2×C6
C3×S3
13C3×S3
13C3×C6
3D26
C3×D13
S3×C13
D39
C3×D13
2C3×D13
3C3×D13
3C78
C3×C39
13S3×C6
S3×D13
3C6×D13
C32×D13
C3×D39
S3×C39
C3×S3×D13

Smallest permutation representation of C3×S3×D13
►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+++++++++
imageC1C2C2C2C3C6C6C6S3D6C3×S3D13S3×C6D26C3×D13C6×D13S3×D13C3×S3×D13
kernelC3×S3×D13C32×D13S3×C39C3×D39S3×D13S3×C13C3×D13D39C3×D13C39D13C3×S3C13C32S3C3C3C1
# reps111122221126261212612

Matrix representation of C3×S3×D13 ►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] >;
 

C3×S3×D13 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 C3×S3×D13 in TeX

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