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## G = C3×He3.2C6order 486 = 2·35

### Direct product of C3 and He3.2C6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — C3×He3.2C6
 Chief series C1 — C3 — C32 — He3 — C3×He3 — C3×He3⋊C3 — C3×He3.2C6
 Lower central He3 — C3×He3.2C6
 Upper central C1 — C32

Generators and relations for C3×He3.2C6
G = < a,b,c,d,e | a3=b3=c3=d3=1, e6=c, 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: 468 in 90 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, C33, C33, S3×C9, He3⋊C2, C3×C18, S3×C32, He3⋊C3, He3⋊C3, C32×C9, C3×He3, C3×He3, He3.2C6, S3×C3×C9, C3×He3⋊C2, C3×He3⋊C3, C3×He3.2C6
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, C32⋊C6, S3×C32, He3.2C6, C3×C32⋊C6, C3×He3.2C6

Smallest permutation representation of C3×He3.2C6
On 81 points
Generators in S81
(1 22 13)(2 23 14)(3 24 15)(4 25 16)(5 26 17)(6 27 18)(7 19 10)(8 20 11)(9 21 12)(28 57 80)(29 58 81)(30 59 64)(31 60 65)(32 61 66)(33 62 67)(34 63 68)(35 46 69)(36 47 70)(37 48 71)(38 49 72)(39 50 73)(40 51 74)(41 52 75)(42 53 76)(43 54 77)(44 55 78)(45 56 79)
(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 34 40)(30 36 42)(32 38 44)(47 53 59)(49 55 61)(51 57 63)(64 70 76)(66 72 78)(68 74 80)
(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)(28 34 40)(29 35 41)(30 36 42)(31 37 43)(32 38 44)(33 39 45)(46 52 58)(47 53 59)(48 54 60)(49 55 61)(50 56 62)(51 57 63)(64 70 76)(65 71 77)(66 72 78)(67 73 79)(68 74 80)(69 75 81)
(1 44 35)(2 42 33)(3 40 31)(4 38 29)(5 36 45)(6 34 43)(7 32 41)(8 30 39)(9 28 37)(10 66 75)(11 64 73)(12 80 71)(13 78 69)(14 76 67)(15 74 65)(16 72 81)(17 70 79)(18 68 77)(19 61 52)(20 59 50)(21 57 48)(22 55 46)(23 53 62)(24 51 60)(25 49 58)(26 47 56)(27 63 54)
(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)| (1,22,13)(2,23,14)(3,24,15)(4,25,16)(5,26,17)(6,27,18)(7,19,10)(8,20,11)(9,21,12)(28,57,80)(29,58,81)(30,59,64)(31,60,65)(32,61,66)(33,62,67)(34,63,68)(35,46,69)(36,47,70)(37,48,71)(38,49,72)(39,50,73)(40,51,74)(41,52,75)(42,53,76)(43,54,77)(44,55,78)(45,56,79), (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)(28,34,40)(30,36,42)(32,38,44)(47,53,59)(49,55,61)(51,57,63)(64,70,76)(66,72,78)(68,74,80), (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)(28,34,40)(29,35,41)(30,36,42)(31,37,43)(32,38,44)(33,39,45)(46,52,58)(47,53,59)(48,54,60)(49,55,61)(50,56,62)(51,57,63)(64,70,76)(65,71,77)(66,72,78)(67,73,79)(68,74,80)(69,75,81), (1,44,35)(2,42,33)(3,40,31)(4,38,29)(5,36,45)(6,34,43)(7,32,41)(8,30,39)(9,28,37)(10,66,75)(11,64,73)(12,80,71)(13,78,69)(14,76,67)(15,74,65)(16,72,81)(17,70,79)(18,68,77)(19,61,52)(20,59,50)(21,57,48)(22,55,46)(23,53,62)(24,51,60)(25,49,58)(26,47,56)(27,63,54), (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( (1,22,13)(2,23,14)(3,24,15)(4,25,16)(5,26,17)(6,27,18)(7,19,10)(8,20,11)(9,21,12)(28,57,80)(29,58,81)(30,59,64)(31,60,65)(32,61,66)(33,62,67)(34,63,68)(35,46,69)(36,47,70)(37,48,71)(38,49,72)(39,50,73)(40,51,74)(41,52,75)(42,53,76)(43,54,77)(44,55,78)(45,56,79), (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)(28,34,40)(30,36,42)(32,38,44)(47,53,59)(49,55,61)(51,57,63)(64,70,76)(66,72,78)(68,74,80), (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)(28,34,40)(29,35,41)(30,36,42)(31,37,43)(32,38,44)(33,39,45)(46,52,58)(47,53,59)(48,54,60)(49,55,61)(50,56,62)(51,57,63)(64,70,76)(65,71,77)(66,72,78)(67,73,79)(68,74,80)(69,75,81), (1,44,35)(2,42,33)(3,40,31)(4,38,29)(5,36,45)(6,34,43)(7,32,41)(8,30,39)(9,28,37)(10,66,75)(11,64,73)(12,80,71)(13,78,69)(14,76,67)(15,74,65)(16,72,81)(17,70,79)(18,68,77)(19,61,52)(20,59,50)(21,57,48)(22,55,46)(23,53,62)(24,51,60)(25,49,58)(26,47,56)(27,63,54), (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([[(1,22,13),(2,23,14),(3,24,15),(4,25,16),(5,26,17),(6,27,18),(7,19,10),(8,20,11),(9,21,12),(28,57,80),(29,58,81),(30,59,64),(31,60,65),(32,61,66),(33,62,67),(34,63,68),(35,46,69),(36,47,70),(37,48,71),(38,49,72),(39,50,73),(40,51,74),(41,52,75),(42,53,76),(43,54,77),(44,55,78),(45,56,79)], [(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),(28,34,40),(30,36,42),(32,38,44),(47,53,59),(49,55,61),(51,57,63),(64,70,76),(66,72,78),(68,74,80)], [(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),(28,34,40),(29,35,41),(30,36,42),(31,37,43),(32,38,44),(33,39,45),(46,52,58),(47,53,59),(48,54,60),(49,55,61),(50,56,62),(51,57,63),(64,70,76),(65,71,77),(66,72,78),(67,73,79),(68,74,80),(69,75,81)], [(1,44,35),(2,42,33),(3,40,31),(4,38,29),(5,36,45),(6,34,43),(7,32,41),(8,30,39),(9,28,37),(10,66,75),(11,64,73),(12,80,71),(13,78,69),(14,76,67),(15,74,65),(16,72,81),(17,70,79),(18,68,77),(19,61,52),(20,59,50),(21,57,48),(22,55,46),(23,53,62),(24,51,60),(25,49,58),(26,47,56),(27,63,54)], [(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 ··· 3T 6A ··· 6H 9A ··· 9R 18A ··· 18R order 1 2 3 ··· 3 3 3 3 3 ··· 3 6 ··· 6 9 ··· 9 18 ··· 18 size 1 9 1 ··· 1 6 6 6 18 ··· 18 9 ··· 9 3 ··· 3 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 He3.2C6 C32⋊C6 C3×C32⋊C6 kernel C3×He3.2C6 C3×He3⋊C3 He3.2C6 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×He3.2C6 in GL5(𝔽19)

 11 0 0 0 0 0 11 0 0 0 0 0 7 0 0 0 0 0 7 0 0 0 0 0 7
,
 1 0 0 0 0 0 1 0 0 0 0 0 11 12 11 0 0 0 7 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 7 0 0 0 0 0 7 0 0 0 0 0 7
,
 18 18 0 0 0 1 0 0 0 0 0 0 7 18 7 0 0 4 12 12 0 0 0 1 0
,
 1 0 0 0 0 18 18 0 0 0 0 0 5 2 2 0 0 0 0 17 0 0 0 17 0

G:=sub<GL(5,GF(19))| [11,0,0,0,0,0,11,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,7],[1,0,0,0,0,0,1,0,0,0,0,0,11,0,0,0,0,12,7,0,0,0,11,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,7],[18,1,0,0,0,18,0,0,0,0,0,0,7,4,0,0,0,18,12,1,0,0,7,12,0],[1,18,0,0,0,0,18,0,0,0,0,0,5,0,0,0,0,2,0,17,0,0,2,17,0] >;

C3×He3.2C6 in GAP, Magma, Sage, TeX

C_3\times {\rm He}_3._2C_6
% in TeX

G:=Group("C3xHe3.2C6");
// GroupNames label

G:=SmallGroup(486,121);
// by ID

G=gap.SmallGroup(486,121);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,867,873,8104,382]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^3=1,e^6=c,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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