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## G = C2×C92⋊C3order 486 = 2·35

### Direct product of C2 and C92⋊C3

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

Aliases: C2×C92⋊C3, C9214C6, (C9×C18)⋊1C3, (C3×C6).1He3, C3.He32C6, C32.1(C2×He3), He3⋊C3.3C6, (C3×C18).16C32, C6.6(He3⋊C3), (C3×C9).17(C3×C6), (C2×C3.He3)⋊1C3, C3.6(C2×He3⋊C3), (C2×He3⋊C3).1C3, SmallGroup(486,85)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C2×C92⋊C3
 Chief series C1 — C3 — C32 — C3×C9 — C92 — C92⋊C3 — C2×C92⋊C3
 Lower central C1 — C3 — C32 — C3×C9 — C2×C92⋊C3
 Upper central C1 — C6 — C3×C6 — C3×C18 — C2×C92⋊C3

Generators and relations for C2×C92⋊C3
G = < a,b,c,d | a2=b9=c9=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b7c2, dcd-1=b3c7 >

Smallest permutation representation of C2×C92⋊C3
On 54 points
Generators in S54
(1 18)(2 16)(3 17)(4 10)(5 11)(6 12)(7 14)(8 15)(9 13)(19 50)(20 51)(21 52)(22 53)(23 54)(24 46)(25 47)(26 48)(27 49)(28 43)(29 44)(30 45)(31 37)(32 38)(33 39)(34 40)(35 41)(36 42)
(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 8 5 2 9 6 3 7 4)(10 18 15 11 16 13 12 17 14)(19 24 20 25 21 26 22 27 23)(28 35 33 31 29 36 34 32 30)(37 44 42 40 38 45 43 41 39)(46 51 47 52 48 53 49 54 50)
(1 24 29)(2 21 32)(3 27 35)(4 22 31)(5 19 34)(6 25 28)(7 20 33)(8 26 36)(9 23 30)(10 53 37)(11 50 40)(12 47 43)(13 54 45)(14 51 39)(15 48 42)(16 52 38)(17 49 41)(18 46 44)

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

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

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

70 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F 6A 6B 6C 6D 6E 6F 9A ··· 9X 9Y 9Z 9AA 9AB 18A ··· 18X 18Y 18Z 18AA 18AB order 1 2 3 3 3 3 3 3 6 6 6 6 6 6 9 ··· 9 9 9 9 9 18 ··· 18 18 18 18 18 size 1 1 1 1 3 3 27 27 1 1 3 3 27 27 3 ··· 3 27 27 27 27 3 ··· 3 27 27 27 27

70 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 3 3 type + + image C1 C2 C3 C3 C3 C6 C6 C6 He3 C2×He3 He3⋊C3 C2×He3⋊C3 C92⋊C3 C2×C92⋊C3 kernel C2×C92⋊C3 C92⋊C3 C9×C18 C2×He3⋊C3 C2×C3.He3 C92 He3⋊C3 C3.He3 C3×C6 C32 C6 C3 C2 C1 # reps 1 1 2 2 4 2 2 4 2 2 6 6 18 18

Matrix representation of C2×C92⋊C3 in GL3(𝔽19) generated by

 18 0 0 0 18 0 0 0 18
,
 7 0 0 0 4 0 0 0 5
,
 6 0 0 0 6 0 0 0 9
,
 0 1 0 0 0 1 1 0 0
G:=sub<GL(3,GF(19))| [18,0,0,0,18,0,0,0,18],[7,0,0,0,4,0,0,0,5],[6,0,0,0,6,0,0,0,9],[0,0,1,1,0,0,0,1,0] >;

C2×C92⋊C3 in GAP, Magma, Sage, TeX

C_2\times C_9^2\rtimes C_3
% in TeX

G:=Group("C2xC9^2:C3");
// GroupNames label

G:=SmallGroup(486,85);
// by ID

G=gap.SmallGroup(486,85);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,224,338,873,453,3250]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^9=c^9=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^7*c^2,d*c*d^-1=b^3*c^7>;
// generators/relations

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