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## G = He3.5C18order 486 = 2·35

### The non-split extension by He3 of C18 acting via C18/C9=C2

Aliases: He3.5C18, (C3×C27)⋊8S3, C27○He32C2, C27.2(C3⋊S3), C9○He3.5C6, C32.5(S3×C9), He3⋊C2.3C9, He3.4C6.3C3, C9.7(C3×C3⋊S3), C3.7(C9×C3⋊S3), (C3×C9).26(C3×S3), SmallGroup(486,164)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — He3.5C18
 Chief series C1 — C3 — C32 — He3 — C9○He3 — C27○He3 — He3.5C18
 Lower central He3 — He3.5C18
 Upper central C1 — C27

Generators and relations for He3.5C18
G = < a,b,c,d | a3=b3=c3=1, d18=b, ab=ba, cac-1=ab-1, dad-1=a-1, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 150 in 62 conjugacy classes, 22 normal (10 characteristic)
C1, C2, C3, C3, S3, C6, C9, C9, C32, C18, C3×S3, C27, C27, C3×C9, He3, 3- 1+2, C54, S3×C9, He3⋊C2, C3×C27, C27⋊C3, C9○He3, S3×C27, He3.4C6, C27○He3, He3.5C18
Quotients: C1, C2, C3, S3, C6, C9, C18, C3×S3, C3⋊S3, S3×C9, C3×C3⋊S3, C9×C3⋊S3, He3.5C18

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

Matrix representation of He3.5C18 in GL3(𝔽109) generated by

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

He3.5C18 in GAP, Magma, Sage, TeX

{\rm He}_3._5C_{18}
% in TeX

G:=Group("He3.5C18");
// GroupNames label

G:=SmallGroup(486,164);
// by ID

G=gap.SmallGroup(486,164);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,43,500,867,3244,382]);
// Polycyclic

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

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