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G = C2×3- 1+4  order 486 = 2·35

Direct product of C2 and 3- 1+4

direct product, metabelian, nilpotent (class 2), monomial, 3-elementary

Aliases: C2×3- 1+4, C6.5C34, C18.2C33, C9○He3⋊7C6, (C3×C18)⋊4C32, (C6×He3).8C3, C9.2(C32×C6), C3.5(C33×C6), He3.17(C3×C6), (C3×He3).21C6, C33.19(C3×C6), (C3×C6).14C33, (C2×He3).7C32, C32.12(C32×C6), (C32×C6).18C32, 3- 1+2⋊7(C3×C6), (C6×3- 1+2)⋊11C3, (C3×3- 1+2)⋊18C6, (C2×3- 1+2)⋊6C32, (C3×C9)⋊8(C3×C6), (C2×C9○He3)⋊3C3, SmallGroup(486,255)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — C2×3- 1+4
Chief series
C1 — C3 — C32 — C33 — C3×He3 — 3- 1+4 — C2×3- 1+4
Lower central
C1 — C3 — C2×3- 1+4
Upper central
C1 — C6 — C2×3- 1+4

Generators and relations for C2×3- 1+4
 G = < a,b,c,d,e,f | a2=b3=d3=e3=f3=1, c3=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, dcd-1=fcf-1=bc=cb, ede-1=bd=db, be=eb, bf=fb, ce=ec, df=fd, ef=fe >

Subgroups: 576 in 460 conjugacy classes, 426 normal (10 characteristic)
C1, C2, C3, C3, C6, C6, C9, C32, C32, C32, C18, C3×C6, C3×C6, C3×C6, C3×C9, He3, 3- 1+2, C33, C3×C18, C2×He3, C2×3- 1+2, C32×C6, C3×He3, C3×3- 1+2, C9○He3, C6×He3, C6×3- 1+2, C2×C9○He3, 3- 1+4, C2×3- 1+4
Quotients: C1, C2, C3, C6, C32, C3×C6, C33, C32×C6, C34, C33×C6, 3- 1+4, C2×3- 1+4

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

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

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

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

166 conjugacy classes

class 1  2 3A3B3C···3AB6A6B6C···6AB9A···9BB18A···18BB
order12333···3666···69···918···18
size11113···3113···33···33···3

166 irreducible representations

dim1111111199
type++
imageC1C2C3C3C3C6C6C63- 1+4C2×3- 1+4
kernelC2×3- 1+43- 1+4C6×He3C6×3- 1+2C2×C9○He3C3×He3C3×3- 1+2C9○He3C2C1
# reps11224542245422

Matrix representation of C2×3- 1+4 ►in GL9(𝔽19)

1800000000
0180000000
0018000000
0001800000
0000180000
0000018000
0000001800
0000000180
0000000018
,
1100000000
0110000000
0011000000
0001100000
0000110000
0000011000
0000001100
0000000110
0000000011
,
1001000000
0001810000
0001801000
0001800100
0001800010
0001800001
1001800000
11101800000
10111800000
,
1100000000
0181000000
0180000000
11800110000
11800011000
11801100000
12180000070
12180000007
12180000700
,
100000000
1110000000
1207000000
000100000
1000110000
1200007000
000000100
1000000110
1200000007
,
100000000
010000000
001000000
1001100000
1000110000
1000011000
1200000700
1200000070
1200000007

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],[11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11],[1,0,0,0,0,0,1,1,1,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,10,18,18,18,18,18,18,18,18,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0],[1,0,0,1,1,1,12,12,12,10,18,18,18,18,18,18,18,18,0,1,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0],[1,1,12,0,1,12,0,1,12,0,11,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,7],[1,0,0,1,1,1,12,12,12,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,7] >;
 

C2×3- 1+4 in GAP, Magma, Sage, TeX 

C_2\times 3_-^{1+4}
 
% in TeX
 
G:=Group("C2xES-(3,2)");
 
// GroupNames label
 
G:=SmallGroup(486,255);
 
// by ID
 
G=gap.SmallGroup(486,255);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,1520,735,3250]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^3=d^3=e^3=f^3=1,c^3=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,d*c*d^-1=f*c*f^-1=b*c=c*b,e*d*e^-1=b*d=d*b,b*e=e*b,b*f=f*b,c*e=e*c,d*f=f*d,e*f=f*e>;
 
// generators/relations
 

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