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G = C3×He3.4C6order 486 = 2·35

Direct product of C3 and He3.4C6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — C3×He3.4C6
 Chief series C1 — C3 — C32 — He3 — C9○He3 — C3×C9○He3 — C3×He3.4C6
 Lower central He3 — C3×He3.4C6
 Upper central C1 — C3×C9

Generators and relations for C3×He3.4C6
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-1, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 504 in 204 conjugacy classes, 46 normal (10 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, C32, C18, C3×S3, C3×C6, C3×C9, C3×C9, C3×C9, He3, He3, 3- 1+2, C33, S3×C9, He3⋊C2, C3×C18, S3×C32, C32×C9, C3×He3, C3×3- 1+2, C9○He3, C9○He3, S3×C3×C9, C3×He3⋊C2, He3.4C6, C3×C9○He3, C3×He3.4C6
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3⋊S3, C3×C6, S3×C32, C3×C3⋊S3, He3.4C6, C32×C3⋊S3, C3×He3.4C6

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

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

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

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

90 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3T 6A ··· 6H 9A ··· 9R 9S ··· 9AP 18A ··· 18R order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 9 1 ··· 1 6 ··· 6 9 ··· 9 1 ··· 1 6 ··· 6 9 ··· 9

90 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 3 type + + + image C1 C2 C3 C3 C6 C6 S3 C3×S3 C3×S3 He3.4C6 kernel C3×He3.4C6 C3×C9○He3 C3×He3⋊C2 He3.4C6 C3×He3 C9○He3 C32×C9 C3×C9 C33 C3 # reps 1 1 2 6 2 6 4 24 8 36

Matrix representation of C3×He3.4C6 in GL5(𝔽19)

 11 0 0 0 0 0 11 0 0 0 0 0 11 0 0 0 0 0 11 0 0 0 0 0 11
,
 0 1 0 0 0 18 18 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0
,
 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
,
 0 1 0 0 0 18 18 0 0 0 0 0 1 0 0 0 0 0 7 0 0 0 0 0 11
,
 11 0 0 0 0 8 8 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 4 0

G:=sub<GL(5,GF(19))| [11,0,0,0,0,0,11,0,0,0,0,0,11,0,0,0,0,0,11,0,0,0,0,0,11],[0,18,0,0,0,1,18,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[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],[0,18,0,0,0,1,18,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,11],[11,8,0,0,0,0,8,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,4,0] >;

C3×He3.4C6 in GAP, Magma, Sage, TeX

C_3\times {\rm He}_3._4C_6
% in TeX

G:=Group("C3xHe3.4C6");
// GroupNames label

G:=SmallGroup(486,235);
// by ID

G=gap.SmallGroup(486,235);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,500,867,3244,382]);
// 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*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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