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## G = He3⋊C18order 486 = 2·35

### The semidirect product of He3 and C18 acting via C18/C3=C6

Aliases: He3⋊C18, C3.5C3≀S3, He3⋊C2⋊C9, He3⋊C91C2, (C32×C9)⋊1S3, (C3×He3).3C6, C32.1(S3×C9), C33.29(C3×S3), C3.6(C32⋊C18), C32.47(C32⋊C6), C3.5(He3.C6), C3.10(He3.2C6), (C3×He3⋊C2).1C3, SmallGroup(486,24)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — He3⋊C18
 Chief series C1 — C3 — C32 — He3 — C3×He3 — He3⋊C9 — He3⋊C18
 Lower central He3 — He3⋊C18
 Upper central C1 — C32

Generators and relations for He3⋊C18
G = < a,b,c,d | a3=b3=c3=d18=1, ab=ba, cac-1=ab-1, dad-1=a-1b, bc=cb, bd=db, dcd-1=a-1c-1 >

Subgroups: 330 in 67 conjugacy classes, 15 normal (all characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, C18, C3×S3, C3×C6, C3×C9, He3, He3, C33, C33, S3×C9, He3⋊C2, C3×C18, S3×C32, C32⋊C9, C32×C9, C3×He3, S3×C3×C9, C3×He3⋊C2, He3⋊C9, He3⋊C18
Quotients: C1, C2, C3, S3, C6, C9, C18, C3×S3, S3×C9, C32⋊C6, C32⋊C18, C3≀S3, He3.C6, He3.2C6, He3⋊C18

Smallest permutation representation of He3⋊C18
On 81 points
Generators in S81
(1 15 27)(2 16 19)(3 17 20)(4 18 21)(5 10 22)(6 11 23)(7 12 24)(8 13 25)(9 14 26)(29 59 71)(31 61 73)(33 63 75)(35 47 77)(37 49 79)(39 51 81)(41 53 65)(43 55 67)(45 57 69)
(1 27 15)(2 19 16)(3 20 17)(4 21 18)(5 22 10)(6 23 11)(7 24 12)(8 25 13)(9 26 14)(28 58 70)(29 59 71)(30 60 72)(31 61 73)(32 62 74)(33 63 75)(34 46 76)(35 47 77)(36 48 78)(37 49 79)(38 50 80)(39 51 81)(40 52 64)(41 53 65)(42 54 66)(43 55 67)(44 56 68)(45 57 69)
(1 79 70)(2 29 38)(3 51 60)(4 73 64)(5 41 32)(6 63 54)(7 67 76)(8 35 44)(9 57 48)(10 65 74)(11 33 42)(12 55 46)(13 77 68)(14 45 36)(15 49 58)(16 71 80)(17 39 30)(18 61 52)(19 59 50)(20 81 72)(21 31 40)(22 53 62)(23 75 66)(24 43 34)(25 47 56)(26 69 78)(27 37 28)
(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)| (1,15,27)(2,16,19)(3,17,20)(4,18,21)(5,10,22)(6,11,23)(7,12,24)(8,13,25)(9,14,26)(29,59,71)(31,61,73)(33,63,75)(35,47,77)(37,49,79)(39,51,81)(41,53,65)(43,55,67)(45,57,69), (1,27,15)(2,19,16)(3,20,17)(4,21,18)(5,22,10)(6,23,11)(7,24,12)(8,25,13)(9,26,14)(28,58,70)(29,59,71)(30,60,72)(31,61,73)(32,62,74)(33,63,75)(34,46,76)(35,47,77)(36,48,78)(37,49,79)(38,50,80)(39,51,81)(40,52,64)(41,53,65)(42,54,66)(43,55,67)(44,56,68)(45,57,69), (1,79,70)(2,29,38)(3,51,60)(4,73,64)(5,41,32)(6,63,54)(7,67,76)(8,35,44)(9,57,48)(10,65,74)(11,33,42)(12,55,46)(13,77,68)(14,45,36)(15,49,58)(16,71,80)(17,39,30)(18,61,52)(19,59,50)(20,81,72)(21,31,40)(22,53,62)(23,75,66)(24,43,34)(25,47,56)(26,69,78)(27,37,28), (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( (1,15,27)(2,16,19)(3,17,20)(4,18,21)(5,10,22)(6,11,23)(7,12,24)(8,13,25)(9,14,26)(29,59,71)(31,61,73)(33,63,75)(35,47,77)(37,49,79)(39,51,81)(41,53,65)(43,55,67)(45,57,69), (1,27,15)(2,19,16)(3,20,17)(4,21,18)(5,22,10)(6,23,11)(7,24,12)(8,25,13)(9,26,14)(28,58,70)(29,59,71)(30,60,72)(31,61,73)(32,62,74)(33,63,75)(34,46,76)(35,47,77)(36,48,78)(37,49,79)(38,50,80)(39,51,81)(40,52,64)(41,53,65)(42,54,66)(43,55,67)(44,56,68)(45,57,69), (1,79,70)(2,29,38)(3,51,60)(4,73,64)(5,41,32)(6,63,54)(7,67,76)(8,35,44)(9,57,48)(10,65,74)(11,33,42)(12,55,46)(13,77,68)(14,45,36)(15,49,58)(16,71,80)(17,39,30)(18,61,52)(19,59,50)(20,81,72)(21,31,40)(22,53,62)(23,75,66)(24,43,34)(25,47,56)(26,69,78)(27,37,28), (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([[(1,15,27),(2,16,19),(3,17,20),(4,18,21),(5,10,22),(6,11,23),(7,12,24),(8,13,25),(9,14,26),(29,59,71),(31,61,73),(33,63,75),(35,47,77),(37,49,79),(39,51,81),(41,53,65),(43,55,67),(45,57,69)], [(1,27,15),(2,19,16),(3,20,17),(4,21,18),(5,22,10),(6,23,11),(7,24,12),(8,25,13),(9,26,14),(28,58,70),(29,59,71),(30,60,72),(31,61,73),(32,62,74),(33,63,75),(34,46,76),(35,47,77),(36,48,78),(37,49,79),(38,50,80),(39,51,81),(40,52,64),(41,53,65),(42,54,66),(43,55,67),(44,56,68),(45,57,69)], [(1,79,70),(2,29,38),(3,51,60),(4,73,64),(5,41,32),(6,63,54),(7,67,76),(8,35,44),(9,57,48),(10,65,74),(11,33,42),(12,55,46),(13,77,68),(14,45,36),(15,49,58),(16,71,80),(17,39,30),(18,61,52),(19,59,50),(20,81,72),(21,31,40),(22,53,62),(23,75,66),(24,43,34),(25,47,56),(26,69,78),(27,37,28)], [(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 3 3 6 6 type + + + + image C1 C2 C3 C6 C9 C18 S3 C3×S3 S3×C9 C3≀S3 He3.C6 He3.2C6 C32⋊C6 C32⋊C18 kernel He3⋊C18 He3⋊C9 C3×He3⋊C2 C3×He3 He3⋊C2 He3 C32×C9 C33 C32 C3 C3 C3 C32 C3 # reps 1 1 2 2 6 6 1 2 6 12 12 12 1 2

Matrix representation of He3⋊C18 in GL5(𝔽19)

 1 0 0 0 0 0 1 0 0 0 0 0 11 0 0 0 0 0 1 0 0 0 0 0 7
,
 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 0 1 0 0 0 0 0 1 0 0 1 0 0
,
 0 6 0 0 0 6 0 0 0 0 0 0 11 0 0 0 0 0 0 7 0 0 0 7 0

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

He3⋊C18 in GAP, Magma, Sage, TeX

{\rm He}_3\rtimes C_{18}
% in TeX

G:=Group("He3:C18");
// GroupNames label

G:=SmallGroup(486,24);
// by ID

G=gap.SmallGroup(486,24);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,43,867,873,8104,382]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^3=d^18=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a^-1*b,b*c=c*b,b*d=d*b,d*c*d^-1=a^-1*c^-1>;
// generators/relations

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