Copied to
clipboard

## G = C2×C9.2He3order 486 = 2·35

### Direct product of C2 and C9.2He3

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C9.2He3
 Chief series C1 — C3 — C32 — C3×C9 — C3×3- 1+2 — C9.2He3 — C2×C9.2He3
 Lower central C1 — C3 — C32 — C2×C9.2He3
 Upper central C1 — C6 — C3×C18 — C2×C9.2He3

Generators and relations for C2×C9.2He3
G = < a,b,c,d,e | a2=b9=c3=d3=e3=1, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b7, bd=db, ebe-1=b4, cd=dc, ece-1=b3cd-1, ede-1=b6d >

Subgroups: 306 in 126 conjugacy classes, 66 normal (20 characteristic)
C1, C2, C3, C3, C6, C6, C9, C9, C32, C32, 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, C3×3- 1+2, C9○He3, C2×C3≀C3, C2×He3.C3, C2×He3⋊C3, C2×C3.He3, C6×3- 1+2, C2×C9○He3, C9.2He3, C2×C9.2He3
Quotients: C1, C2, C3, C6, C32, C3×C6, He3, C33, C2×He3, C32×C6, C3×He3, C6×He3, C9.2He3, C2×C9.2He3

Smallest permutation representation of C2×C9.2He3
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)
(2 5 8)(3 9 6)(10 16 13)(12 15 18)(20 23 26)(21 27 24)(28 34 31)(30 33 36)(37 43 40)(39 42 45)(47 50 53)(48 54 51)
(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), (2,5,8)(3,9,6)(10,16,13)(12,15,18)(20,23,26)(21,27,24)(28,34,31)(30,33,36)(37,43,40)(39,42,45)(47,50,53)(48,54,51), (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), (2,5,8)(3,9,6)(10,16,13)(12,15,18)(20,23,26)(21,27,24)(28,34,31)(30,33,36)(37,43,40)(39,42,45)(47,50,53)(48,54,51), (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)], [(2,5,8),(3,9,6),(10,16,13),(12,15,18),(20,23,26),(21,27,24),(28,34,31),(30,33,36),(37,43,40),(39,42,45),(47,50,53),(48,54,51)], [(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)]])

70 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E ··· 3L 6A 6B 6C 6D 6E ··· 6L 9A ··· 9F 9G ··· 9V 18A ··· 18F 18G ··· 18V order 1 2 3 3 3 3 3 ··· 3 6 6 6 6 6 ··· 6 9 ··· 9 9 ··· 9 18 ··· 18 18 ··· 18 size 1 1 1 1 3 3 9 ··· 9 1 1 3 3 9 ··· 9 3 ··· 3 9 ··· 9 3 ··· 3 9 ··· 9

70 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 3 9 9 type + + image C1 C2 C3 C3 C3 C3 C3 C3 C6 C6 C6 C6 C6 C6 He3 C2×He3 C9.2He3 C2×C9.2He3 kernel C2×C9.2He3 C9.2He3 C2×C3≀C3 C2×He3.C3 C2×He3⋊C3 C2×C3.He3 C6×3- 1+2 C2×C9○He3 C3≀C3 He3.C3 He3⋊C3 C3.He3 C3×3- 1+2 C9○He3 C18 C9 C2 C1 # reps 1 1 6 6 2 4 2 6 6 6 2 4 2 6 6 6 2 2

Matrix representation of C2×C9.2He3 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
,
 18 1 11 7 1 11 7 0 18 8 0 7 8 11 6 8 12 18 13 0 1 0 7 8 0 11 18 0 0 0 7 1 11 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 10 8 12 0 0 0 0 0 0 0 0 0 1 11 7 0 0 0 0 0 0 0 0 7 0 0 0 0 0 0 15 12 18
,
 1 0 1 0 18 8 0 0 11 0 11 18 0 0 12 0 8 0 0 0 7 0 0 0 0 0 0 0 0 0 1 11 8 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 1 18 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 11
,
 1 0 0 0 7 8 0 12 11 0 1 0 0 8 1 0 11 18 0 0 1 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 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 12 0 1 0 0 0 0 0 0 11 0 0 0 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 1 11 0 0 0 0 0 0 0 0 10 7 0 12 11 0 8 1 0 0 1 0 0 0 0 0 0

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],[18,8,13,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,11,7,1,0,0,0,0,0,0,7,8,0,7,0,10,0,0,0,1,11,7,1,0,8,0,0,0,11,6,8,11,11,12,0,0,0,7,8,0,0,0,0,1,0,15,0,12,11,0,0,0,11,0,12,18,18,18,0,0,0,7,7,18],[1,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,1,18,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,18,0,0,11,11,0,0,0,0,8,12,0,8,0,7,0,0,0,0,0,0,0,0,0,7,0,0,0,8,0,0,0,0,1,1,0,11,0,0,0,0,0,18,0,11],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,7,8,0,0,7,0,0,0,0,8,1,0,0,0,7,0,0,0,0,0,0,0,0,0,11,0,0,12,11,0,0,0,0,0,11,0,11,18,0,0,0,0,0,0,11],[0,0,0,0,0,0,1,0,0,12,11,0,0,0,0,11,10,0,0,0,0,0,0,0,0,7,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,1,0,0,0,0,11,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,0,0,0,0,1,0,1,0] >;

C2×C9.2He3 in GAP, Magma, Sage, TeX

C_2\times C_9._2{\rm He}_3
% in TeX

G:=Group("C2xC9.2He3");
// GroupNames label

G:=SmallGroup(486,219);
// by ID

G=gap.SmallGroup(486,219);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,548,735,237,3250]);
// 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,c*b*c^-1=b^7,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

׿
×
𝔽