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## G = C3×He3.C6order 486 = 2·35

### Direct product of C3 and He3.C6

Aliases: C3×He3.C6, (C32×C9)⋊4S3, He3.C39C6, He3.1(C3×C6), (C3×He3).5C6, C33.35(C3×S3), C32.2(S3×C32), He3⋊C2.1C32, C32.50(C32⋊C6), (C3×C9)⋊10(C3×S3), (C3×He3.C3)⋊5C2, C3.16(C3×C32⋊C6), (C3×He3⋊C2).2C3, SmallGroup(486,118)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — C3×He3.C6
 Chief series C1 — C3 — C32 — He3 — C3×He3 — C3×He3.C3 — C3×He3.C6
 Lower central He3 — C3×He3.C6
 Upper central C1 — C32

Generators and relations for C3×He3.C6
G = < a,b,c,d,e | a3=b3=c3=d3=1, e6=c-1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, ebe-1=b-1c, cd=dc, ce=ec, ede-1=b-1d-1 >

Subgroups: 360 in 84 conjugacy classes, 24 normal (12 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C32, C32, C18, C3×S3, C3×C6, C3×C9, C3×C9, He3, He3, 3- 1+2, C33, C33, S3×C9, He3⋊C2, C3×C18, S3×C32, He3.C3, He3.C3, C32×C9, C3×He3, C3×3- 1+2, He3.C6, S3×C3×C9, C3×He3⋊C2, C3×He3.C3, C3×He3.C6
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, C32⋊C6, S3×C32, He3.C6, C3×C32⋊C6, C3×He3.C6

Smallest permutation representation of C3×He3.C6
On 81 points
Generators in S81
(1 25 11)(2 26 12)(3 27 13)(4 19 14)(5 20 15)(6 21 16)(7 22 17)(8 23 18)(9 24 10)(28 47 67)(29 48 68)(30 49 69)(31 50 70)(32 51 71)(33 52 72)(34 53 73)(35 54 74)(36 55 75)(37 56 76)(38 57 77)(39 58 78)(40 59 79)(41 60 80)(42 61 81)(43 62 64)(44 63 65)(45 46 66)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(29 41 35)(31 43 37)(33 45 39)(46 58 52)(48 60 54)(50 62 56)(64 76 70)(66 78 72)(68 80 74)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 40 34)(29 41 35)(30 42 36)(31 43 37)(32 44 38)(33 45 39)(46 58 52)(47 59 53)(48 60 54)(49 61 55)(50 62 56)(51 63 57)(64 76 70)(65 77 71)(66 78 72)(67 79 73)(68 80 74)(69 81 75)
(1 72 81)(2 76 67)(3 80 71)(4 66 75)(5 70 79)(6 74 65)(7 78 69)(8 64 73)(9 68 77)(10 48 57)(11 52 61)(12 56 47)(13 60 51)(14 46 55)(15 50 59)(16 54 63)(17 58 49)(18 62 53)(19 45 36)(20 31 40)(21 35 44)(22 39 30)(23 43 34)(24 29 38)(25 33 42)(26 37 28)(27 41 32)
(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)

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

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

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

66 conjugacy classes

 class 1 2 3A ··· 3H 3I 3J 3K 3L 3M 3N 6A ··· 6H 9A ··· 9R 9S ··· 9X 18A ··· 18R order 1 2 3 ··· 3 3 3 3 3 3 3 6 ··· 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 9 1 ··· 1 6 6 6 18 18 18 9 ··· 9 3 ··· 3 18 ··· 18 9 ··· 9

66 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 3 6 6 type + + + + image C1 C2 C3 C3 C6 C6 S3 C3×S3 C3×S3 He3.C6 C32⋊C6 C3×C32⋊C6 kernel C3×He3.C6 C3×He3.C3 He3.C6 C3×He3⋊C2 He3.C3 C3×He3 C32×C9 C3×C9 C33 C3 C32 C3 # reps 1 1 6 2 6 2 1 6 2 36 1 2

Matrix representation of C3×He3.C6 in GL5(𝔽19)

 7 0 0 0 0 0 7 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 7 18 7 0 0 0 1 0 0 0 0 0 11
,
 1 0 0 0 0 0 1 0 0 0 0 0 11 0 0 0 0 0 11 0 0 0 0 0 11
,
 18 1 0 0 0 18 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 6 18 18
,
 0 1 0 0 0 1 0 0 0 0 0 0 17 17 17 0 0 0 0 5 0 0 0 5 0

G:=sub<GL(5,GF(19))| [7,0,0,0,0,0,7,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,18,1,0,0,0,7,0,11],[1,0,0,0,0,0,1,0,0,0,0,0,11,0,0,0,0,0,11,0,0,0,0,0,11],[18,18,0,0,0,1,0,0,0,0,0,0,1,0,6,0,0,0,0,18,0,0,0,1,18],[0,1,0,0,0,1,0,0,0,0,0,0,17,0,0,0,0,17,0,5,0,0,17,5,0] >;

C3×He3.C6 in GAP, Magma, Sage, TeX

C_3\times {\rm He}_3.C_6
% in TeX

G:=Group("C3xHe3.C6");
// GroupNames label

G:=SmallGroup(486,118);
// by ID

G=gap.SmallGroup(486,118);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,867,873,12964,652]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^3=1,e^6=c^-1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b*c^-1,e*b*e^-1=b^-1*c,c*d=d*c,c*e=e*c,e*d*e^-1=b^-1*d^-1>;
// generators/relations

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