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G = C2×C9.He3order 486 = 2·35

Direct product of C2 and C9.He3

direct product, metabelian, nilpotent (class 3), monomial, 3-elementary

Aliases: C2×C9.He3, C18.6He3, C3≀C39C6, C9○He35C6, C9.6(C2×He3), (C32×C18)⋊8C3, (C32×C9)⋊34C6, He3.9(C3×C6), (C3×C6).5C33, C6.11(C3×He3), C3.11(C6×He3), He3.C311C6, C3.He38C6, C33.46(C3×C6), He3⋊C313C6, (C3×C18).22C32, (C2×He3).2C32, C32.5(C32×C6), (C32×C6).34C32, 3- 1+2.2(C3×C6), (C2×3- 1+2).2C32, (C2×C3≀C3)⋊3C3, (C3×C9).31(C3×C6), (C2×C9○He3)⋊1C3, (C2×He3.C3)⋊3C3, (C2×He3⋊C3)⋊5C3, (C2×C3.He3)⋊6C3, SmallGroup(486,214)

Series: Derived Chief Lower central Upper central

C1C32 — C2×C9.He3
C1C3C32C3×C9C32×C9C9.He3 — C2×C9.He3
C1C3C32 — C2×C9.He3
C1C18C3×C18 — C2×C9.He3

Generators and relations for C2×C9.He3
 G = < a,b,c,d,e | a2=b9=c3=d3=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b4, cd=dc, ece-1=b3cd-1, ede-1=b6d >

Subgroups: 306 in 134 conjugacy classes, 66 normal (22 characteristic)
C1, C2, C3, C3, C6, C6, C9, C9, C9, C32, C32, C18, C18, C18, C3×C6, C3×C6, C3×C9, C3×C9, He3, 3- 1+2, 3- 1+2, C33, C3×C18, C3×C18, C2×He3, C2×3- 1+2, C2×3- 1+2, C32×C6, C3≀C3, He3.C3, He3⋊C3, C3.He3, C32×C9, C9○He3, C2×C3≀C3, C2×He3.C3, C2×He3⋊C3, C2×C3.He3, C32×C18, C2×C9○He3, C9.He3, C2×C9.He3
Quotients: C1, C2, C3, C6, C32, C3×C6, He3, C33, C2×He3, C32×C6, C3×He3, C6×He3, C9.He3, C2×C9.He3

Smallest permutation representation of C2×C9.He3
On 54 points
Generators in S54
(1 35)(2 36)(3 28)(4 29)(5 30)(6 31)(7 32)(8 33)(9 34)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 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)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)
(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)(37 40 43)(38 41 44)(39 42 45)(46 52 49)(47 53 50)(48 54 51)
(1 20 14)(2 27 18)(3 25 13)(4 23 17)(5 21 12)(6 19 16)(7 26 11)(8 24 15)(9 22 10)(28 52 40)(29 50 44)(30 48 39)(31 46 43)(32 53 38)(33 51 42)(34 49 37)(35 47 41)(36 54 45)

G:=sub<Sym(54)| (1,35)(2,36)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54), (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), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42), (10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24)(37,40,43)(38,41,44)(39,42,45)(46,52,49)(47,53,50)(48,54,51), (1,20,14)(2,27,18)(3,25,13)(4,23,17)(5,21,12)(6,19,16)(7,26,11)(8,24,15)(9,22,10)(28,52,40)(29,50,44)(30,48,39)(31,46,43)(32,53,38)(33,51,42)(34,49,37)(35,47,41)(36,54,45)>;

G:=Group( (1,35)(2,36)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54), (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), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42), (10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24)(37,40,43)(38,41,44)(39,42,45)(46,52,49)(47,53,50)(48,54,51), (1,20,14)(2,27,18)(3,25,13)(4,23,17)(5,21,12)(6,19,16)(7,26,11)(8,24,15)(9,22,10)(28,52,40)(29,50,44)(30,48,39)(31,46,43)(32,53,38)(33,51,42)(34,49,37)(35,47,41)(36,54,45) );

G=PermutationGroup([[(1,35),(2,36),(3,28),(4,29),(5,30),(6,31),(7,32),(8,33),(9,34),(10,37),(11,38),(12,39),(13,40),(14,41),(15,42),(16,43),(17,44),(18,45),(19,46),(20,47),(21,48),(22,49),(23,50),(24,51),(25,52),(26,53),(27,54)], [(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)], [(1,7,4),(2,8,5),(3,9,6),(10,16,13),(11,17,14),(12,18,15),(28,34,31),(29,35,32),(30,36,33),(37,43,40),(38,44,41),(39,45,42)], [(10,13,16),(11,14,17),(12,15,18),(19,25,22),(20,26,23),(21,27,24),(37,40,43),(38,41,44),(39,42,45),(46,52,49),(47,53,50),(48,54,51)], [(1,20,14),(2,27,18),(3,25,13),(4,23,17),(5,21,12),(6,19,16),(7,26,11),(8,24,15),(9,22,10),(28,52,40),(29,50,44),(30,48,39),(31,46,43),(32,53,38),(33,51,42),(34,49,37),(35,47,41),(36,54,45)]])

102 conjugacy classes

class 1  2 3A3B3C···3J3K···3P6A6B6C···6J6K···6P9A···9F9G···9V9W···9AH18A···18F18G···18V18W···18AH
order12333···33···3666···66···69···99···99···918···1818···1818···18
size11113···39···9113···39···91···13···39···91···13···39···9

102 irreducible representations

dim111111111111113333
type++
imageC1C2C3C3C3C3C3C3C6C6C6C6C6C6He3C2×He3C9.He3C2×C9.He3
kernelC2×C9.He3C9.He3C2×C3≀C3C2×He3.C3C2×He3⋊C3C2×C3.He3C32×C18C2×C9○He3C3≀C3He3.C3He3⋊C3C3.He3C32×C9C9○He3C18C9C2C1
# reps11662426662426661818

Matrix representation of C2×C9.He3 in GL4(𝔽19) generated by

18000
0100
0010
0001
,
7000
0400
0069
0009
,
7000
01100
00118
0001
,
1000
0100
00712
00011
,
7000
00116
01102
01000
G:=sub<GL(4,GF(19))| [18,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[7,0,0,0,0,4,0,0,0,0,6,0,0,0,9,9],[7,0,0,0,0,11,0,0,0,0,11,0,0,0,8,1],[1,0,0,0,0,1,0,0,0,0,7,0,0,0,12,11],[7,0,0,0,0,0,11,10,0,1,0,0,0,16,2,0] >;

C2×C9.He3 in GAP, Magma, Sage, TeX

C_2\times C_9.{\rm He}_3
% in TeX

G:=Group("C2xC9.He3");
// GroupNames label

G:=SmallGroup(486,214);
// by ID

G=gap.SmallGroup(486,214);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,548,237,3250]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^9=c^3=d^3=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b^4,c*d=d*c,e*c*e^-1=b^3*c*d^-1,e*d*e^-1=b^6*d>;
// generators/relations

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