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G = C2×C32.5He3  order 486 = 2·35

Direct product of C2 and C32.5He3

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

Aliases: C2×C32.5He3, C9⋊C9.2C6, (C3×C6).5He3, (C3×C18).3C32, C32.5(C2×He3), C3.He3.5C6, C6.10(He3⋊C3), (C2×C9⋊C9).2C3, (C3×C9).3(C3×C6), C3.10(C2×He3⋊C3), (C2×C3.He3).2C3, SmallGroup(486,89)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C9 — C2×C32.5He3
Chief series
C1 — C3 — C32 — C3×C9 — C9⋊C9 — C32.5He3 — C2×C32.5He3
Lower central
C1 — C3 — C32 — C3×C9 — C2×C32.5He3
Upper central
C1 — C6 — C3×C6 — C3×C18 — C2×C32.5He3

Generators and relations for C2×C32.5He3
 G = < a,b,c,d,e,f | a2=b3=c3=1, d3=b-1, e3=c-1, f3=c, 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
C6
3C6
C32
3C9
9C9
9C9
9C9
9C9
C3×C6
3C18
9C18
9C18
9C18
9C18
C3×C9
33- 1+2
33- 1+2
33- 1+2
3C3×C9
C3×C18
3C2×3- 1+2
3C2×3- 1+2
3C2×3- 1+2
3C3×C18
C3.He3
C9⋊C9
C3.He3
C3.He3
C2×C3.He3
C2×C9⋊C9
C2×C3.He3
C2×C3.He3
C32.5He3
C2×C32.5He3

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

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

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

38 conjugacy classes

class 1  2 3A3B3C3D6A6B6C6D9A···9H9I···9N18A···18H18I···18N
order12333366669···99···918···1818···18
size11113311339···927···279···927···27

38 irreducible representations

dim111111333399
type++
imageC1C2C3C3C6C6He3C2×He3He3⋊C3C2×He3⋊C3C32.5He3C2×C32.5He3
kernelC2×C32.5He3C32.5He3C2×C9⋊C9C2×C3.He3C9⋊C9C3.He3C3×C6C32C6C3C2C1
# reps112626226622

Matrix representation of C2×C32.5He3 ►in GL12(𝔽19)

1800000000000
0180000000000
0018000000000
000100000000
000010000000
000001000000
000000100000
000000010000
000000001000
000000000100
000000000010
000000000001
,
1100000000000
0110000000000
0011000000000
000100000000
000010000000
000001000000
000000700000
000000070000
000777007000
0000000001100
0000000000110
0001212120000011
,
100000000000
010000000000
001000000000
000700000000
000070000000
000007000000
000000700000
000000070000
000000007000
000000000700
000000000070
000000000007
,
600000000000
060000000000
009000000000
000100000000
0000110000000
000007000000
0000000110000
00011111112124000
000780777000
00011100012124
0000000001100
00011180000707
,
1100000000000
010000000000
007000000000
000010000000
000001000000
0001100000000
000000010000
00077718186000
000100801000
000000000070
00011100012124
0001811110001807
,
007000000000
700000000000
070000000000
000000100000
000000010000
00077718186000
000000000100
000000000010
0000000012001
000700000000
000070000000
000111008000

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],[11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,7,0,0,12,0,0,0,0,1,0,0,0,7,0,0,12,0,0,0,0,0,1,0,0,7,0,0,12,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,11],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,7],[6,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,11,7,1,0,11,0,0,0,0,11,0,0,11,8,1,0,18,0,0,0,0,0,7,0,11,0,1,0,0,0,0,0,0,0,0,0,12,7,0,0,0,0,0,0,0,0,0,11,12,7,0,0,0,0,0,0,0,0,0,0,4,7,0,0,0,0,0,0,0,0,0,0,0,0,12,11,7,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,4,0,7],[11,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,7,1,0,1,18,0,0,0,1,0,0,0,7,0,0,1,11,0,0,0,0,1,0,0,7,0,0,1,11,0,0,0,0,0,0,0,18,8,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,6,1,0,0,0,0,0,0,0,0,0,0,0,0,0,12,18,0,0,0,0,0,0,0,0,0,7,12,0,0,0,0,0,0,0,0,0,0,0,4,7],[0,7,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,7,0,1,0,0,0,0,0,7,0,0,0,0,7,1,0,0,0,0,0,7,0,0,0,0,0,1,0,0,0,1,0,18,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,6,0,0,12,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0] >;
 

C2×C32.5He3 in GAP, Magma, Sage, TeX 

C_2\times C_3^2._5{\rm He}_3
 
% in TeX
 
G:=Group("C2xC3^2.5He3");
 
// GroupNames label
 
G:=SmallGroup(486,89);
 
// by ID
 
G=gap.SmallGroup(486,89);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,224,338,873,735,453,3250]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=1,d^3=b^-1,e^3=c^-1,f^3=c,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×C32.5He3 in TeX

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