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## G = S32×C13order 468 = 22·32·13

### Direct product of C13, S3 and S3

Aliases: S32×C13, C395D6, C3⋊S3⋊C26, (C3×S3)⋊C26, C32⋊(C2×C26), C31(S3×C26), (S3×C39)⋊3C2, (C3×C39)⋊5C22, (C13×C3⋊S3)⋊3C2, SmallGroup(468,44)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — S32×C13
 Chief series C1 — C3 — C32 — C3×C39 — S3×C39 — S32×C13
 Lower central C32 — S32×C13
 Upper central C1 — C13

Generators and relations for S32×C13
G = < a,b,c,d,e | a13=b3=c2=d3=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 S32×C13
On 78 points
Generators in S78
(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 21 59)(2 22 60)(3 23 61)(4 24 62)(5 25 63)(6 26 64)(7 14 65)(8 15 53)(9 16 54)(10 17 55)(11 18 56)(12 19 57)(13 20 58)(27 42 68)(28 43 69)(29 44 70)(30 45 71)(31 46 72)(32 47 73)(33 48 74)(34 49 75)(35 50 76)(36 51 77)(37 52 78)(38 40 66)(39 41 67)
(1 52)(2 40)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 51)(14 30)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 27)(25 28)(26 29)(53 72)(54 73)(55 74)(56 75)(57 76)(58 77)(59 78)(60 66)(61 67)(62 68)(63 69)(64 70)(65 71)
(1 59 21)(2 60 22)(3 61 23)(4 62 24)(5 63 25)(6 64 26)(7 65 14)(8 53 15)(9 54 16)(10 55 17)(11 56 18)(12 57 19)(13 58 20)(27 42 68)(28 43 69)(29 44 70)(30 45 71)(31 46 72)(32 47 73)(33 48 74)(34 49 75)(35 50 76)(36 51 77)(37 52 78)(38 40 66)(39 41 67)
(1 52)(2 40)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 51)(14 71)(15 72)(16 73)(17 74)(18 75)(19 76)(20 77)(21 78)(22 66)(23 67)(24 68)(25 69)(26 70)(27 62)(28 63)(29 64)(30 65)(31 53)(32 54)(33 55)(34 56)(35 57)(36 58)(37 59)(38 60)(39 61)

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

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

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

117 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 6A 6B 13A ··· 13L 26A ··· 26X 26Y ··· 26AJ 39A ··· 39X 39Y ··· 39AJ 78A ··· 78X order 1 2 2 2 3 3 3 6 6 13 ··· 13 26 ··· 26 26 ··· 26 39 ··· 39 39 ··· 39 78 ··· 78 size 1 3 3 9 2 2 4 6 6 1 ··· 1 3 ··· 3 9 ··· 9 2 ··· 2 4 ··· 4 6 ··· 6

117 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + image C1 C2 C2 C13 C26 C26 S3 D6 S3×C13 S3×C26 S32 S32×C13 kernel S32×C13 S3×C39 C13×C3⋊S3 S32 C3×S3 C3⋊S3 S3×C13 C39 S3 C3 C13 C1 # reps 1 2 1 12 24 12 2 2 24 24 1 12

Matrix representation of S32×C13 in GL4(𝔽79) generated by

 64 0 0 0 0 64 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 78 1 0 0 78 0
,
 78 0 0 0 0 78 0 0 0 0 0 1 0 0 1 0
,
 0 1 0 0 78 78 0 0 0 0 1 0 0 0 0 1
,
 78 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(79))| [64,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,78,78,0,0,1,0],[78,0,0,0,0,78,0,0,0,0,0,1,0,0,1,0],[0,78,0,0,1,78,0,0,0,0,1,0,0,0,0,1],[78,1,0,0,0,1,0,0,0,0,1,0,0,0,0,1] >;

S32×C13 in GAP, Magma, Sage, TeX

S_3^2\times C_{13}
% in TeX

G:=Group("S3^2xC13");
// GroupNames label

G:=SmallGroup(468,44);
// by ID

G=gap.SmallGroup(468,44);
# by ID

G:=PCGroup([5,-2,-2,-13,-3,-3,1048,7804]);
// Polycyclic

G:=Group<a,b,c,d,e|a^13=b^3=c^2=d^3=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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