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## G = C3×S3×D13order 468 = 22·32·13

### Direct product of C3, S3 and D13

Aliases: C3×S3×D13, C394D6, D393C6, C323D26, C134(S3×C6), C395(C2×C6), C31(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 >

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 2A 2B 2C 3A 3B 3C 3D 3E 6A 6B 6C 6D 6E 6F 6G 6H 6I 13A ··· 13F 26A ··· 26F 39A ··· 39L 39M ··· 39AD 78A ··· 78L order 1 2 2 2 3 3 3 3 3 6 6 6 6 6 6 6 6 6 13 ··· 13 26 ··· 26 39 ··· 39 39 ··· 39 78 ··· 78 size 1 3 13 39 1 1 2 2 2 3 3 13 13 26 26 26 39 39 2 ··· 2 6 ··· 6 2 ··· 2 4 ··· 4 6 ··· 6

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D6 C3×S3 D13 S3×C6 D26 C3×D13 C6×D13 S3×D13 C3×S3×D13 kernel C3×S3×D13 C32×D13 S3×C39 C3×D39 S3×D13 S3×C13 C3×D13 D39 C3×D13 C39 D13 C3×S3 C13 C32 S3 C3 C3 C1 # reps 1 1 1 1 2 2 2 2 1 1 2 6 2 6 12 12 6 12

Matrix representation of C3×S3×D13 in GL4(𝔽79) generated by

 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 1 0 0 78 0
,
 1 0 0 0 0 1 0 0 0 0 25 67 0 0 52 54
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

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