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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, C6.4C34, (C6×He3)⋊7C3, He3⋊7(C3×C6), C33⋊6(C3×C6), (C3×He3)⋊19C6, C3.4(C33×C6), (C3×C6).13C33, (C2×He3)⋊4C32, (C32×C6)⋊3C32, C32.11(C32×C6), SmallGroup(486,254)

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,g | a2=b3=c3=d3=e3=g3=1, f1=c-1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, dbd-1=bc-1, be=eb, bf=fb, bg=gb, cd=dc, geg-1=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, ef=fe, fg=gf >

Subgroups: 1386 in 586 conjugacy classes, 426 normal (6 characteristic)
C1, C2, C3, C3, C6, C6, C32, C32, C3×C6, C3×C6, He3, C33, C2×He3, C32×C6, C3×He3, C6×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 34)(2 35)(3 36)(4 29)(5 30)(6 28)(7 31)(8 32)(9 33)(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 15 10)(2 13 11)(3 14 12)(4 9 53)(5 7 54)(6 8 52)(16 20 23)(17 21 24)(18 19 22)(25 28 32)(26 29 33)(27 30 31)(34 42 37)(35 40 38)(36 41 39)(43 47 50)(44 48 51)(45 46 49)

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

(4 9 53)(5 7 54)(6 8 52)(16 23 20)(17 24 21)(18 22 19)(25 28 32)(26 29 33)(27 30 31)(43 50 47)(44 51 48)(45 49 46)

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

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

166 conjugacy classes

class 1  2 3A3B3C···3CD6A6B6C···6CD
order12333···3666···6
size11113···3113···3

166 irreducible representations

dim111199
type++
imageC1C2C3C63+ 1+4C2×3+ 1+4
kernelC2×3+ 1+43+ 1+4C6×He3C3×He3C2C1
# reps11808022

Matrix representation of C2×3+ 1+4 ►in GL10(𝔽7)

6000000000
0100000000
0010000000
0001000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
1000000000
0010000000
0362000000
0251000000
0000020000
0000654000
0000432000
0000000040
0000000531
0000000164
,
1000000000
0400000000
0040000000
0004000000
0000400000
0000040000
0000004000
0000000400
0000000040
0000000004
,
2000000000
0531000000
0100000000
0002000000
0000531000
0000100000
0000002000
0000000531
0000000100
0000000002
,
4000000000
0100000000
0010000000
0001000000
0000200000
0000020000
0000002000
0000000400
0000000040
0000000004
,
1000000000
0200000000
0020000000
0002000000
0000200000
0000020000
0000002000
0000000200
0000000020
0000000002
,
1000000000
0000531000
0000100000
0000002000
0000000654
0000000400
0000000001
0362000000
0200000000
0004000000

G:=sub<GL(10,GF(7))| [6,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,0,3,2,0,0,0,0,0,0,0,1,6,5,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,0,0,6,4,0,0,0,0,0,0,0,2,5,3,0,0,0,0,0,0,0,0,4,2,0,0,0,0,0,0,0,0,0,0,0,5,1,0,0,0,0,0,0,0,4,3,6,0,0,0,0,0,0,0,0,1,4],[1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,0,0,0,0,5,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,1,0,2,0,0,0,0,0,0,0,0,0,0,5,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,1,0,2,0,0,0,0,0,0,0,0,0,0,5,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,1,0,2],[4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,2],[1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,2,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,2,0,4,0,5,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,1,0,2,0,0,0,0,0,0,0,0,0,0,6,4,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,4,0,1,0,0,0] >;
 

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(486,254);
 
// by ID
 
G=gap.SmallGroup(486,254);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1520,735,3250]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^3=c^3=d^3=e^3=g^3=1,f^1=c^-1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,d*b*d^-1=b*c^-1,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,g*e*g^-1=c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,e*f=f*e,f*g=g*f>;
 
// generators/relations
 

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