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## G = C32×He3⋊C2order 486 = 2·35

### Direct product of C32 and He3⋊C2

Aliases: C32×He3⋊C2, C3410S3, He36(C3×C6), (C3×He3)⋊23C6, C3310(C3×S3), (C32×He3)⋊4C2, C322(S3×C32), C33.53(C3⋊S3), C3.8(C32×C3⋊S3), C32.57(C3×C3⋊S3), SmallGroup(486,230)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — C32×He3⋊C2
 Chief series C1 — C3 — C32 — He3 — C3×He3 — C32×He3 — C32×He3⋊C2
 Lower central He3 — C32×He3⋊C2
 Upper central C1 — C33

Generators and relations for C32×He3⋊C2
G = < a,b,c,d,e,f | a3=b3=c3=d3=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ece-1=cd-1, fcf=c-1, de=ed, df=fd, fef=e-1 >

Subgroups: 1476 in 444 conjugacy classes, 64 normal (7 characteristic)
C1, C2, C3, C3, C3, S3, C6, C32, C32, C3×S3, C3×C6, He3, He3, C33, C33, C33, He3⋊C2, S3×C32, C32×C6, C3×He3, C3×He3, C34, C3×He3⋊C2, S3×C33, C32×He3, C32×He3⋊C2
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3⋊S3, C3×C6, He3⋊C2, S3×C32, C3×C3⋊S3, C3×He3⋊C2, C32×C3⋊S3, C32×He3⋊C2

Smallest permutation representation of C32×He3⋊C2
On 81 points
Generators in S81
(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)(79 80 81)
(1 80 77)(2 81 78)(3 79 76)(4 62 59)(5 63 60)(6 61 58)(7 73 10)(8 74 11)(9 75 12)(13 19 16)(14 20 17)(15 21 18)(22 29 25)(23 30 26)(24 28 27)(31 37 34)(32 38 35)(33 39 36)(40 46 43)(41 47 44)(42 48 45)(49 56 52)(50 57 53)(51 55 54)(64 70 67)(65 71 68)(66 72 69)
(1 49 24)(2 50 22)(3 51 23)(4 7 33)(5 8 31)(6 9 32)(10 36 59)(11 34 60)(12 35 58)(13 66 41)(14 64 42)(15 65 40)(16 69 44)(17 67 45)(18 68 43)(19 72 47)(20 70 48)(21 71 46)(25 78 53)(26 76 54)(27 77 52)(28 80 56)(29 81 57)(30 79 55)(37 63 74)(38 61 75)(39 62 73)
(1 5 64)(2 6 65)(3 4 66)(7 41 51)(8 42 49)(9 40 50)(10 44 54)(11 45 52)(12 43 53)(13 23 33)(14 24 31)(15 22 32)(16 26 36)(17 27 34)(18 25 35)(19 30 39)(20 28 37)(21 29 38)(46 57 75)(47 55 73)(48 56 74)(58 68 78)(59 69 76)(60 67 77)(61 71 81)(62 72 79)(63 70 80)
(1 42 14)(2 40 15)(3 41 13)(4 51 23)(5 49 24)(6 50 22)(7 33 66)(8 31 64)(9 32 65)(10 36 69)(11 34 67)(12 35 68)(16 76 44)(17 77 45)(18 78 43)(19 79 47)(20 80 48)(21 81 46)(25 58 53)(26 59 54)(27 60 52)(28 63 56)(29 61 57)(30 62 55)(37 70 74)(38 71 75)(39 72 73)
(7 33)(8 31)(9 32)(10 36)(11 34)(12 35)(13 41)(14 42)(15 40)(16 44)(17 45)(18 43)(19 47)(20 48)(21 46)(22 50)(23 51)(24 49)(25 53)(26 54)(27 52)(28 56)(29 57)(30 55)(37 74)(38 75)(39 73)

G:=sub<Sym(81)| (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)(79,80,81), (1,80,77)(2,81,78)(3,79,76)(4,62,59)(5,63,60)(6,61,58)(7,73,10)(8,74,11)(9,75,12)(13,19,16)(14,20,17)(15,21,18)(22,29,25)(23,30,26)(24,28,27)(31,37,34)(32,38,35)(33,39,36)(40,46,43)(41,47,44)(42,48,45)(49,56,52)(50,57,53)(51,55,54)(64,70,67)(65,71,68)(66,72,69), (1,49,24)(2,50,22)(3,51,23)(4,7,33)(5,8,31)(6,9,32)(10,36,59)(11,34,60)(12,35,58)(13,66,41)(14,64,42)(15,65,40)(16,69,44)(17,67,45)(18,68,43)(19,72,47)(20,70,48)(21,71,46)(25,78,53)(26,76,54)(27,77,52)(28,80,56)(29,81,57)(30,79,55)(37,63,74)(38,61,75)(39,62,73), (1,5,64)(2,6,65)(3,4,66)(7,41,51)(8,42,49)(9,40,50)(10,44,54)(11,45,52)(12,43,53)(13,23,33)(14,24,31)(15,22,32)(16,26,36)(17,27,34)(18,25,35)(19,30,39)(20,28,37)(21,29,38)(46,57,75)(47,55,73)(48,56,74)(58,68,78)(59,69,76)(60,67,77)(61,71,81)(62,72,79)(63,70,80), (1,42,14)(2,40,15)(3,41,13)(4,51,23)(5,49,24)(6,50,22)(7,33,66)(8,31,64)(9,32,65)(10,36,69)(11,34,67)(12,35,68)(16,76,44)(17,77,45)(18,78,43)(19,79,47)(20,80,48)(21,81,46)(25,58,53)(26,59,54)(27,60,52)(28,63,56)(29,61,57)(30,62,55)(37,70,74)(38,71,75)(39,72,73), (7,33)(8,31)(9,32)(10,36)(11,34)(12,35)(13,41)(14,42)(15,40)(16,44)(17,45)(18,43)(19,47)(20,48)(21,46)(22,50)(23,51)(24,49)(25,53)(26,54)(27,52)(28,56)(29,57)(30,55)(37,74)(38,75)(39,73)>;

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)(79,80,81), (1,80,77)(2,81,78)(3,79,76)(4,62,59)(5,63,60)(6,61,58)(7,73,10)(8,74,11)(9,75,12)(13,19,16)(14,20,17)(15,21,18)(22,29,25)(23,30,26)(24,28,27)(31,37,34)(32,38,35)(33,39,36)(40,46,43)(41,47,44)(42,48,45)(49,56,52)(50,57,53)(51,55,54)(64,70,67)(65,71,68)(66,72,69), (1,49,24)(2,50,22)(3,51,23)(4,7,33)(5,8,31)(6,9,32)(10,36,59)(11,34,60)(12,35,58)(13,66,41)(14,64,42)(15,65,40)(16,69,44)(17,67,45)(18,68,43)(19,72,47)(20,70,48)(21,71,46)(25,78,53)(26,76,54)(27,77,52)(28,80,56)(29,81,57)(30,79,55)(37,63,74)(38,61,75)(39,62,73), (1,5,64)(2,6,65)(3,4,66)(7,41,51)(8,42,49)(9,40,50)(10,44,54)(11,45,52)(12,43,53)(13,23,33)(14,24,31)(15,22,32)(16,26,36)(17,27,34)(18,25,35)(19,30,39)(20,28,37)(21,29,38)(46,57,75)(47,55,73)(48,56,74)(58,68,78)(59,69,76)(60,67,77)(61,71,81)(62,72,79)(63,70,80), (1,42,14)(2,40,15)(3,41,13)(4,51,23)(5,49,24)(6,50,22)(7,33,66)(8,31,64)(9,32,65)(10,36,69)(11,34,67)(12,35,68)(16,76,44)(17,77,45)(18,78,43)(19,79,47)(20,80,48)(21,81,46)(25,58,53)(26,59,54)(27,60,52)(28,63,56)(29,61,57)(30,62,55)(37,70,74)(38,71,75)(39,72,73), (7,33)(8,31)(9,32)(10,36)(11,34)(12,35)(13,41)(14,42)(15,40)(16,44)(17,45)(18,43)(19,47)(20,48)(21,46)(22,50)(23,51)(24,49)(25,53)(26,54)(27,52)(28,56)(29,57)(30,55)(37,74)(38,75)(39,73) );

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),(79,80,81)], [(1,80,77),(2,81,78),(3,79,76),(4,62,59),(5,63,60),(6,61,58),(7,73,10),(8,74,11),(9,75,12),(13,19,16),(14,20,17),(15,21,18),(22,29,25),(23,30,26),(24,28,27),(31,37,34),(32,38,35),(33,39,36),(40,46,43),(41,47,44),(42,48,45),(49,56,52),(50,57,53),(51,55,54),(64,70,67),(65,71,68),(66,72,69)], [(1,49,24),(2,50,22),(3,51,23),(4,7,33),(5,8,31),(6,9,32),(10,36,59),(11,34,60),(12,35,58),(13,66,41),(14,64,42),(15,65,40),(16,69,44),(17,67,45),(18,68,43),(19,72,47),(20,70,48),(21,71,46),(25,78,53),(26,76,54),(27,77,52),(28,80,56),(29,81,57),(30,79,55),(37,63,74),(38,61,75),(39,62,73)], [(1,5,64),(2,6,65),(3,4,66),(7,41,51),(8,42,49),(9,40,50),(10,44,54),(11,45,52),(12,43,53),(13,23,33),(14,24,31),(15,22,32),(16,26,36),(17,27,34),(18,25,35),(19,30,39),(20,28,37),(21,29,38),(46,57,75),(47,55,73),(48,56,74),(58,68,78),(59,69,76),(60,67,77),(61,71,81),(62,72,79),(63,70,80)], [(1,42,14),(2,40,15),(3,41,13),(4,51,23),(5,49,24),(6,50,22),(7,33,66),(8,31,64),(9,32,65),(10,36,69),(11,34,67),(12,35,68),(16,76,44),(17,77,45),(18,78,43),(19,79,47),(20,80,48),(21,81,46),(25,58,53),(26,59,54),(27,60,52),(28,63,56),(29,61,57),(30,62,55),(37,70,74),(38,71,75),(39,72,73)], [(7,33),(8,31),(9,32),(10,36),(11,34),(12,35),(13,41),(14,42),(15,40),(16,44),(17,45),(18,43),(19,47),(20,48),(21,46),(22,50),(23,51),(24,49),(25,53),(26,54),(27,52),(28,56),(29,57),(30,55),(37,74),(38,75),(39,73)]])

90 conjugacy classes

 class 1 2 3A ··· 3Z 3AA ··· 3BJ 6A ··· 6Z order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 size 1 9 1 ··· 1 6 ··· 6 9 ··· 9

90 irreducible representations

 dim 1 1 1 1 2 2 3 type + + + image C1 C2 C3 C6 S3 C3×S3 He3⋊C2 kernel C32×He3⋊C2 C32×He3 C3×He3⋊C2 C3×He3 C34 C33 C32 # reps 1 1 8 8 4 32 36

Matrix representation of C32×He3⋊C2 in GL7(𝔽7)

 4 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 2
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 4
,
 6 1 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 6 6 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0
,
 1 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 0 0 0 0 0 0 2 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 2
,
 0 6 0 0 0 0 0 1 6 0 0 0 0 0 0 0 6 6 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 1 0 0 0 0 4 0 0
,
 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0

G:=sub<GL(7,GF(7))| [4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4],[6,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,1,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2],[0,1,0,0,0,0,0,6,6,0,0,0,0,0,0,0,6,1,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,2,0,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0] >;

C32×He3⋊C2 in GAP, Magma, Sage, TeX

C_3^2\times {\rm He}_3\rtimes C_2
% in TeX

G:=Group("C3^2xHe3:C2");
// GroupNames label

G:=SmallGroup(486,230);
// by ID

G=gap.SmallGroup(486,230);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,867,3244,382]);
// Polycyclic

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

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