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## G = C2×3+ 1+4order 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, He37(C3×C6), C336(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 3A 3B 3C ··· 3CD 6A 6B 6C ··· 6CD order 1 2 3 3 3 ··· 3 6 6 6 ··· 6 size 1 1 1 1 3 ··· 3 1 1 3 ··· 3

166 irreducible representations

 dim 1 1 1 1 9 9 type + + image C1 C2 C3 C6 3+ 1+4 C2×3+ 1+4 kernel C2×3+ 1+4 3+ 1+4 C6×He3 C3×He3 C2 C1 # reps 1 1 80 80 2 2

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

 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 1 0 0 0 0 0 0 0 0 3 6 2 0 0 0 0 0 0 0 2 5 1 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 6 5 4 0 0 0 0 0 0 0 4 3 2 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 5 3 1 0 0 0 0 0 0 0 1 6 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 3 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 5 3 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 5 3 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 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 5 3 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 6 5 4 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 1 0 3 6 2 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0

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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