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G = C2×C9.2He3  order 486 = 2·35

Direct product of C2 and C9.2He3

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

Aliases: C2×C9.2He3, C18.He3, C3≀C3⋊5C6, C9.(C2×He3), C9○He3⋊6C6, He3.C3⋊6C6, C3.16(C6×He3), C6.16(C3×He3), He3⋊C3⋊8C6, He3.12(C3×C6), (C3×C6).10C33, C3.He3⋊6C6, C33.18(C3×C6), (C3×C18).15C32, (C2×He3).5C32, (C32×C6).17C32, C32.10(C32×C6), (C3×3- 1+2)⋊17C6, (C6×3- 1+2)⋊10C3, 3- 1+2.4(C3×C6), (C2×3- 1+2).4C32, (C2×C3≀C3)⋊2C3, (C3×C9).9(C3×C6), (C2×C9○He3)⋊2C3, (C2×He3.C3)⋊2C3, (C2×He3⋊C3)⋊4C3, (C2×C3.He3)⋊5C3, SmallGroup(486,219)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C2×C9.2He3
Chief series
C1 — C3 — C32 — C3×C9 — C3×3- 1+2 — C9.2He3 — C2×C9.2He3
Lower central
C1 — C3 — C32 — C2×C9.2He3
Upper central
C1 — C6 — C3×C18 — C2×C9.2He3

Generators and relations for C2×C9.2He3
 G = < a,b,c,d,e | a2=b9=c3=d3=e3=1, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b7, bd=db, ebe-1=b4, cd=dc, ece-1=b3cd-1, ede-1=b6d >

Subgroups: 306 in 126 conjugacy classes, 66 normal (20 characteristic)
C1, C2, C3, C3, C6, C6, C9, C9, C32, C32, 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, C3×3- 1+2, C9○He3, C2×C3≀C3, C2×He3.C3, C2×He3⋊C3, C2×C3.He3, C6×3- 1+2, C2×C9○He3, C9.2He3, C2×C9.2He3
Quotients: C1, C2, C3, C6, C32, C3×C6, He3, C33, C2×He3, C32×C6, C3×He3, C6×He3, C9.2He3, C2×C9.2He3

Smallest permutation representation of C2×C9.2He3
►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)

(2 5 8)(3 9 6)(10 16 13)(12 15 18)(20 23 26)(21 27 24)(28 34 31)(30 33 36)(37 43 40)(39 42 45)(47 50 53)(48 54 51)

(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), (2,5,8)(3,9,6)(10,16,13)(12,15,18)(20,23,26)(21,27,24)(28,34,31)(30,33,36)(37,43,40)(39,42,45)(47,50,53)(48,54,51), (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), (2,5,8)(3,9,6)(10,16,13)(12,15,18)(20,23,26)(21,27,24)(28,34,31)(30,33,36)(37,43,40)(39,42,45)(47,50,53)(48,54,51), (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)], [(2,5,8),(3,9,6),(10,16,13),(12,15,18),(20,23,26),(21,27,24),(28,34,31),(30,33,36),(37,43,40),(39,42,45),(47,50,53),(48,54,51)], [(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)]])
 

70 conjugacy classes

class 1  2 3A3B3C3D3E···3L6A6B6C6D6E···6L9A···9F9G···9V18A···18F18G···18V
order1233333···366666···69···99···918···1818···18
size1111339···911339···93···39···93···39···9

70 irreducible representations

dim111111111111113399
type++
imageC1C2C3C3C3C3C3C3C6C6C6C6C6C6He3C2×He3C9.2He3C2×C9.2He3
kernelC2×C9.2He3C9.2He3C2×C3≀C3C2×He3.C3C2×He3⋊C3C2×C3.He3C6×3- 1+2C2×C9○He3C3≀C3He3.C3He3⋊C3C3.He3C3×3- 1+2C9○He3C18C9C2C1
# reps116624266624266622

Matrix representation of C2×C9.2He3 ►in GL9(𝔽19)

1800000000
0180000000
0018000000
0001800000
0000180000
0000018000
0000001800
0000000180
0000000018
,
1811171117018
807811681218
130107801118
0007111000
0000011000
00010812000
0000001117
000000007
000000151218
,
10101880011
011180012080
007000000
0001118000
0000110000
000007000
0000007118
000000010
0000000011
,
10007801211
01008101118
001000000
000700000
000070000
000007000
0000001100
0000000110
0000000011
,
0120100000
0110000000
000001000
000000100
000000010
000000001
1110000000
010701211081
001000000

G:=sub<GL(9,GF(19))| [18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18],[18,8,13,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,11,7,1,0,0,0,0,0,0,7,8,0,7,0,10,0,0,0,1,11,7,1,0,8,0,0,0,11,6,8,11,11,12,0,0,0,7,8,0,0,0,0,1,0,15,0,12,11,0,0,0,11,0,12,18,18,18,0,0,0,7,7,18],[1,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,1,18,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,18,0,0,11,11,0,0,0,0,8,12,0,8,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,8,0,0,0,0,1,1,0,11,0,0,0,0,0,18,0,11],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,7,8,0,0,7,0,0,0,0,8,1,0,0,0,7,0,0,0,0,0,0,0,0,0,11,0,0,12,11,0,0,0,0,0,11,0,11,18,0,0,0,0,0,0,11],[0,0,0,0,0,0,1,0,0,12,11,0,0,0,0,11,10,0,0,0,0,0,0,0,0,7,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,1,0,0,0,0,11,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,0,0,0,0,1,0,1,0] >;
 

C2×C9.2He3 in GAP, Magma, Sage, TeX 

C_2\times C_9._2{\rm He}_3
 
% in TeX
 
G:=Group("C2xC9.2He3");
 
// GroupNames label
 
G:=SmallGroup(486,219);
 
// by ID
 
G=gap.SmallGroup(486,219);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,548,735,237,3250]);
 
// 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,c*b*c^-1=b^7,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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