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G = C2×C32.6He3  order 486 = 2·35

Direct product of C2 and C32.6He3

direct product, metabelian, nilpotent (class 4), monomial, 3-elementary

Aliases: C2×C32.6He3, C9⋊C9⋊5C6, (C3×C6).6He3, (C3×C18).4C32, C3.He3⋊4C6, C32.6(C2×He3), He3⋊C3.4C6, C6.11(He3⋊C3), (C2×C9⋊C9)⋊2C3, (C3×C9).4(C3×C6), (C2×C3.He3)⋊3C3, (C2×He3⋊C3).2C3, C3.11(C2×He3⋊C3), SmallGroup(486,90)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C9 — C2×C32.6He3
Chief series
C1 — C3 — C32 — C3×C9 — C9⋊C9 — C32.6He3 — C2×C32.6He3
Lower central
C1 — C3 — C32 — C3×C9 — C2×C32.6He3
Upper central
C1 — C6 — C3×C6 — C3×C18 — C2×C32.6He3

Generators and relations for C2×C32.6He3
 G = < a,b,c,d,e,f | a2=b3=c3=1, d3=b-1, e3=f3=c-1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=bc-1, ede-1=cd=dc, ce=ec, cf=fc, fdf-1=bde2, fef-1=b-1ce >

C1
C2
C3
3C3
27C3
C6
3C6
27C6
C32
3C9
9C32
9C9
9C9
9C9
C3×C6
3C18
9C18
9C18
9C18
9C3×C6
C3×C9
3He3
33- 1+2
33- 1+2
3C3×C9
C3×C18
3C2×He3
3C2×3- 1+2
3C2×3- 1+2
3C3×C18
C3.He3
C9⋊C9
C3.He3
He3⋊C3
C2×C3.He3
C2×C9⋊C9
C2×C3.He3
C2×He3⋊C3
C32.6He3
C2×C32.6He3

Smallest permutation representation of C2×C32.6He3
►On 54 points
Generators in S54
(1 5)(2 4)(3 6)(7 14)(8 15)(9 13)(10 17)(11 18)(12 16)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 38)(29 39)(30 40)(31 41)(32 42)(33 43)(34 44)(35 45)(36 37)

(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 52 49)(47 53 50)(48 54 51)

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

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

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

38 conjugacy classes

class 1  2 3A3B3C3D3E3F6A6B6C6D6E6F9A···9H9I9J9K9L18A···18H18I18J18K18L
order123333336666669···9999918···1818181818
size1111332727113327279···9272727279···927272727

38 irreducible representations

dim11111111333399
type++
imageC1C2C3C3C3C6C6C6He3C2×He3He3⋊C3C2×He3⋊C3C32.6He3C2×C32.6He3
kernelC2×C32.6He3C32.6He3C2×C9⋊C9C2×He3⋊C3C2×C3.He3C9⋊C9He3⋊C3C3.He3C3×C6C32C6C3C2C1
# reps11224224226622

Matrix representation of C2×C32.6He3 ►in GL12(𝔽19)

1800000000000
0180000000000
0018000000000
000100000000
000010000000
000001000000
000000100000
000000010000
000000001000
000000000100
000000000010
000000000001
,
700000000000
070000000000
007000000000
000100000000
000010000000
000001000000
0000001100000
0000000110000
0000000011000
000000000700
000000000070
000000000007
,
100000000000
010000000000
001000000000
0001100000000
0000110000000
0000011000000
0000001100000
0000000110000
0000000011000
0000000001100
0000000000110
0000000000011
,
500000000000
15167000000000
005000000000
000100000000
000070000000
0000011000000
000000070000
0000000011000
000000700000
0000000000011
000000000700
0000000000110
,
700000000000
21115000000000
001000000000
000010000000
000001000000
000700000000
000000010000
000000001000
000000700000
0000000000110
0000000000011
000000000100
,
161310000000000
1039000000000
700000000000
000000100000
000000010000
000000001000
000000000100
000000000010
000000000001
000700000000
000070000000
000007000000

G:=sub<GL(12,GF(19))| [18,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1],[7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11],[5,15,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0,7,5,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0],[7,2,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0],[16,10,7,0,0,0,0,0,0,0,0,0,13,3,0,0,0,0,0,0,0,0,0,0,10,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0] >;
 

C2×C32.6He3 in GAP, Magma, Sage, TeX 

C_2\times C_3^2._6{\rm He}_3
 
% in TeX
 
G:=Group("C2xC3^2.6He3");
 
// GroupNames label
 
G:=SmallGroup(486,90);
 
// by ID
 
G=gap.SmallGroup(486,90);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1951,224,338,873,735,453,3250]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=1,d^3=b^-1,e^3=f^3=c^-1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=b*c^-1,e*d*e^-1=c*d=d*c,c*e=e*c,c*f=f*c,f*d*f^-1=b*d*e^2,f*e*f^-1=b^-1*c*e>;
 
// generators/relations
 

Export

Subgroup lattice of C2×C32.6He3 in TeX

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