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## G = C2×C9.He3order 486 = 2·35

### Direct product of C2 and C9.He3

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C9.He3
 Chief series C1 — C3 — C32 — C3×C9 — C32×C9 — C9.He3 — C2×C9.He3
 Lower central C1 — C3 — C32 — C2×C9.He3
 Upper central C1 — C18 — C3×C18 — C2×C9.He3

Generators and relations for C2×C9.He3
G = < a,b,c,d,e | a2=b9=c3=d3=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b4, cd=dc, ece-1=b3cd-1, ede-1=b6d >

Subgroups: 306 in 134 conjugacy classes, 66 normal (22 characteristic)
C1, C2, C3, C3, C6, C6, C9, C9, C9, C32, C32, C18, C18, C18, C3×C6, C3×C6, C3×C9, C3×C9, He3, 3- 1+2, 3- 1+2, C33, C3×C18, C3×C18, C2×He3, C2×3- 1+2, C2×3- 1+2, C32×C6, C3≀C3, He3.C3, He3⋊C3, C3.He3, C32×C9, C9○He3, C2×C3≀C3, C2×He3.C3, C2×He3⋊C3, C2×C3.He3, C32×C18, C2×C9○He3, C9.He3, C2×C9.He3
Quotients: C1, C2, C3, C6, C32, C3×C6, He3, C33, C2×He3, C32×C6, C3×He3, C6×He3, C9.He3, C2×C9.He3

Smallest permutation representation of C2×C9.He3
On 54 points
Generators in S54
(1 35)(2 36)(3 28)(4 29)(5 30)(6 31)(7 32)(8 33)(9 34)(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 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)
(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)(37 40 43)(38 41 44)(39 42 45)(46 52 49)(47 53 50)(48 54 51)
(1 20 14)(2 27 18)(3 25 13)(4 23 17)(5 21 12)(6 19 16)(7 26 11)(8 24 15)(9 22 10)(28 52 40)(29 50 44)(30 48 39)(31 46 43)(32 53 38)(33 51 42)(34 49 37)(35 47 41)(36 54 45)

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

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

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

102 conjugacy classes

 class 1 2 3A 3B 3C ··· 3J 3K ··· 3P 6A 6B 6C ··· 6J 6K ··· 6P 9A ··· 9F 9G ··· 9V 9W ··· 9AH 18A ··· 18F 18G ··· 18V 18W ··· 18AH 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 9 ··· 9 1 1 3 ··· 3 9 ··· 9 1 ··· 1 3 ··· 3 9 ··· 9 1 ··· 1 3 ··· 3 9 ··· 9

102 irreducible representations

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

Matrix representation of C2×C9.He3 in GL4(𝔽19) generated by

 18 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 7 0 0 0 0 4 0 0 0 0 6 9 0 0 0 9
,
 7 0 0 0 0 11 0 0 0 0 11 8 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 7 12 0 0 0 11
,
 7 0 0 0 0 0 1 16 0 11 0 2 0 10 0 0
G:=sub<GL(4,GF(19))| [18,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[7,0,0,0,0,4,0,0,0,0,6,0,0,0,9,9],[7,0,0,0,0,11,0,0,0,0,11,0,0,0,8,1],[1,0,0,0,0,1,0,0,0,0,7,0,0,0,12,11],[7,0,0,0,0,0,11,10,0,1,0,0,0,16,2,0] >;

C2×C9.He3 in GAP, Magma, Sage, TeX

C_2\times C_9.{\rm He}_3
% in TeX

G:=Group("C2xC9.He3");
// GroupNames label

G:=SmallGroup(486,214);
// by ID

G=gap.SmallGroup(486,214);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,548,237,3250]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^9=c^3=d^3=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b^4,c*d=d*c,e*c*e^-1=b^3*c*d^-1,e*d*e^-1=b^6*d>;
// generators/relations

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