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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

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 3A 3B 3C ··· 3AB 6A 6B 6C ··· 6AB 9A ··· 9BB 18A ··· 18BB order 1 2 3 3 3 ··· 3 6 6 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 1 1 3 ··· 3 1 1 3 ··· 3 3 ··· 3 3 ··· 3

166 irreducible representations

 dim 1 1 1 1 1 1 1 1 9 9 type + + image C1 C2 C3 C3 C3 C6 C6 C6 3- 1+4 C2×3- 1+4 kernel C2×3- 1+4 3- 1+4 C6×He3 C6×3- 1+2 C2×C9○He3 C3×He3 C3×3- 1+2 C9○He3 C2 C1 # reps 1 1 2 24 54 2 24 54 2 2

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

 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 10 0 0 0 0 0 0 0 0 18 1 0 0 0 0 0 0 0 18 0 1 0 0 0 0 0 0 18 0 0 1 0 0 0 0 0 18 0 0 0 1 0 0 0 0 18 0 0 0 0 1 1 0 0 18 0 0 0 0 0 1 11 0 18 0 0 0 0 0 1 0 11 18 0 0 0 0 0
,
 1 10 0 0 0 0 0 0 0 0 18 1 0 0 0 0 0 0 0 18 0 0 0 0 0 0 0 1 18 0 0 11 0 0 0 0 1 18 0 0 0 11 0 0 0 1 18 0 11 0 0 0 0 0 12 18 0 0 0 0 0 7 0 12 18 0 0 0 0 0 0 7 12 18 0 0 0 0 7 0 0
,
 1 0 0 0 0 0 0 0 0 1 11 0 0 0 0 0 0 0 12 0 7 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 11 0 0 0 0 12 0 0 0 0 7 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 11 0 12 0 0 0 0 0 0 0 7
,
 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 1 0 0 11 0 0 0 0 0 1 0 0 0 11 0 0 0 0 1 0 0 0 0 11 0 0 0 12 0 0 0 0 0 7 0 0 12 0 0 0 0 0 0 7 0 12 0 0 0 0 0 0 0 7

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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