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## G = He3.2C18order 486 = 2·35

### 2nd non-split extension by He3 of C18 acting via C18/C3=C6

Aliases: He3.2C18, (C3×C27)⋊3S3, C9○He3.2C6, C32.3(S3×C9), C9.6He33C2, C9.8(C32⋊C6), He3⋊C2.2C9, C3.8(C32⋊C18), He3.4C6.2C3, (C3×C9).24(C3×S3), SmallGroup(486,28)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — He3.2C18
 Chief series C1 — C3 — C32 — He3 — C9○He3 — C9.6He3 — He3.2C18
 Lower central He3 — He3.2C18
 Upper central C1 — C9

Generators and relations for He3.2C18
G = < a,b,c,d | a3=b3=c3=1, d18=b-1, ab=ba, cac-1=ab-1, dad-1=a-1b, bc=cb, bd=db, dcd-1=a-1c-1 >

Smallest permutation representation of He3.2C18
On 81 points
Generators in S81
(1 19 10)(2 20 11)(3 21 12)(4 22 13)(5 23 14)(6 24 15)(7 25 16)(8 26 17)(9 27 18)(29 65 47)(31 67 49)(33 69 51)(35 71 53)(37 73 55)(39 75 57)(41 77 59)(43 79 61)(45 81 63)
(1 10 19)(2 11 20)(3 12 21)(4 13 22)(5 14 23)(6 15 24)(7 16 25)(8 17 26)(9 18 27)(28 64 46)(29 65 47)(30 66 48)(31 67 49)(32 68 50)(33 69 51)(34 70 52)(35 71 53)(36 72 54)(37 73 55)(38 74 56)(39 75 57)(40 76 58)(41 77 59)(42 78 60)(43 79 61)(44 80 62)(45 81 63)
(1 45 72)(2 55 28)(3 65 38)(4 75 48)(5 31 58)(6 41 68)(7 51 78)(8 61 34)(9 71 44)(10 81 54)(11 37 64)(12 47 74)(13 57 30)(14 67 40)(15 77 50)(16 33 60)(17 43 70)(18 53 80)(19 63 36)(20 73 46)(21 29 56)(22 39 66)(23 49 76)(24 59 32)(25 69 42)(26 79 52)(27 35 62)
(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)| (1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(29,65,47)(31,67,49)(33,69,51)(35,71,53)(37,73,55)(39,75,57)(41,77,59)(43,79,61)(45,81,63), (1,10,19)(2,11,20)(3,12,21)(4,13,22)(5,14,23)(6,15,24)(7,16,25)(8,17,26)(9,18,27)(28,64,46)(29,65,47)(30,66,48)(31,67,49)(32,68,50)(33,69,51)(34,70,52)(35,71,53)(36,72,54)(37,73,55)(38,74,56)(39,75,57)(40,76,58)(41,77,59)(42,78,60)(43,79,61)(44,80,62)(45,81,63), (1,45,72)(2,55,28)(3,65,38)(4,75,48)(5,31,58)(6,41,68)(7,51,78)(8,61,34)(9,71,44)(10,81,54)(11,37,64)(12,47,74)(13,57,30)(14,67,40)(15,77,50)(16,33,60)(17,43,70)(18,53,80)(19,63,36)(20,73,46)(21,29,56)(22,39,66)(23,49,76)(24,59,32)(25,69,42)(26,79,52)(27,35,62), (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( (1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(29,65,47)(31,67,49)(33,69,51)(35,71,53)(37,73,55)(39,75,57)(41,77,59)(43,79,61)(45,81,63), (1,10,19)(2,11,20)(3,12,21)(4,13,22)(5,14,23)(6,15,24)(7,16,25)(8,17,26)(9,18,27)(28,64,46)(29,65,47)(30,66,48)(31,67,49)(32,68,50)(33,69,51)(34,70,52)(35,71,53)(36,72,54)(37,73,55)(38,74,56)(39,75,57)(40,76,58)(41,77,59)(42,78,60)(43,79,61)(44,80,62)(45,81,63), (1,45,72)(2,55,28)(3,65,38)(4,75,48)(5,31,58)(6,41,68)(7,51,78)(8,61,34)(9,71,44)(10,81,54)(11,37,64)(12,47,74)(13,57,30)(14,67,40)(15,77,50)(16,33,60)(17,43,70)(18,53,80)(19,63,36)(20,73,46)(21,29,56)(22,39,66)(23,49,76)(24,59,32)(25,69,42)(26,79,52)(27,35,62), (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([[(1,19,10),(2,20,11),(3,21,12),(4,22,13),(5,23,14),(6,24,15),(7,25,16),(8,26,17),(9,27,18),(29,65,47),(31,67,49),(33,69,51),(35,71,53),(37,73,55),(39,75,57),(41,77,59),(43,79,61),(45,81,63)], [(1,10,19),(2,11,20),(3,12,21),(4,13,22),(5,14,23),(6,15,24),(7,16,25),(8,17,26),(9,18,27),(28,64,46),(29,65,47),(30,66,48),(31,67,49),(32,68,50),(33,69,51),(34,70,52),(35,71,53),(36,72,54),(37,73,55),(38,74,56),(39,75,57),(40,76,58),(41,77,59),(42,78,60),(43,79,61),(44,80,62),(45,81,63)], [(1,45,72),(2,55,28),(3,65,38),(4,75,48),(5,31,58),(6,41,68),(7,51,78),(8,61,34),(9,71,44),(10,81,54),(11,37,64),(12,47,74),(13,57,30),(14,67,40),(15,77,50),(16,33,60),(17,43,70),(18,53,80),(19,63,36),(20,73,46),(21,29,56),(22,39,66),(23,49,76),(24,59,32),(25,69,42),(26,79,52),(27,35,62)], [(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 3B 3C 3D 6A 6B 9A ··· 9F 9G 9H 9I 9J 18A ··· 18F 27A ··· 27R 27S ··· 27X 54A ··· 54R order 1 2 3 3 3 3 6 6 9 ··· 9 9 9 9 9 18 ··· 18 27 ··· 27 27 ··· 27 54 ··· 54 size 1 9 1 1 6 18 9 9 1 ··· 1 6 6 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 C6 C9 C18 S3 C3×S3 S3×C9 He3.2C18 C32⋊C6 C32⋊C18 kernel He3.2C18 C9.6He3 He3.4C6 C9○He3 He3⋊C2 He3 C3×C27 C3×C9 C32 C1 C9 C3 # reps 1 1 2 2 6 6 1 2 6 36 1 2

Matrix representation of He3.2C18 in GL3(𝔽109) generated by

 45 0 0 0 1 0 0 0 63
,
 63 0 0 0 63 0 0 0 63
,
 0 1 0 0 0 1 1 0 0
,
 5 0 0 0 0 7 0 7 0
G:=sub<GL(3,GF(109))| [45,0,0,0,1,0,0,0,63],[63,0,0,0,63,0,0,0,63],[0,0,1,1,0,0,0,1,0],[5,0,0,0,0,7,0,7,0] >;

He3.2C18 in GAP, Magma, Sage, TeX

{\rm He}_3._2C_{18}
% in TeX

G:=Group("He3.2C18");
// GroupNames label

G:=SmallGroup(486,28);
// by ID

G=gap.SmallGroup(486,28);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,43,500,867,873,12964,652]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^3=1,d^18=b^-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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