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G = C2xC9oHe3order 162 = 2·34

Direct product of C2 and C9oHe3

direct product, metabelian, nilpotent (class 2), monomial, 3-elementary

Aliases: C2xC9oHe3, C18.C32, He3.5C6, C6.3C33, 3- 1+2:4C6, (C3xC9):12C6, (C3xC18):5C3, C9.2(C3xC6), (C2xHe3).2C3, C32.5(C3xC6), (C3xC6).5C32, C3.3(C32xC6), (C2x3- 1+2):3C3, SmallGroup(162,50)

Series: Derived Chief Lower central Upper central

C1C3 — C2xC9oHe3
C1C3C9C3xC9C9oHe3 — C2xC9oHe3
C1C3 — C2xC9oHe3
C1C18 — C2xC9oHe3

Generators and relations for C2xC9oHe3
 G = < a,b,c,d,e | a2=b9=c3=e3=1, d1=b6, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=b3c, de=ed >

Subgroups: 82 in 66 conjugacy classes, 58 normal (10 characteristic)
C1, C2, C3, C3, C6, C6, C9, C9, C32, C18, C18, C3xC6, C3xC9, He3, 3- 1+2, C3xC18, C2xHe3, C2x3- 1+2, C9oHe3, C2xC9oHe3
Quotients: C1, C2, C3, C6, C32, C3xC6, C33, C32xC6, C9oHe3, C2xC9oHe3

Smallest permutation representation of C2xC9oHe3
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)
(10 16 13)(11 17 14)(12 18 15)(19 22 25)(20 23 26)(21 24 27)(37 43 40)(38 44 41)(39 45 42)(46 49 52)(47 50 53)(48 51 54)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 52 49)(47 53 50)(48 54 51)
(1 26 11)(2 27 12)(3 19 13)(4 20 14)(5 21 15)(6 22 16)(7 23 17)(8 24 18)(9 25 10)(28 46 40)(29 47 41)(30 48 42)(31 49 43)(32 50 44)(33 51 45)(34 52 37)(35 53 38)(36 54 39)

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

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

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

C2xC9oHe3 is a maximal subgroup of   He3.4Dic3  He3.5C12
C2xC9oHe3 is a maximal quotient of   C18xHe3  C18x3- 1+2

66 conjugacy classes

class 1  2 3A3B3C···3J6A6B6C···6J9A···9F9G···9V18A···18F18G···18V
order12333···3666···69···99···918···1818···18
size11113···3113···31···13···31···13···3

66 irreducible representations

dim1111111133
type++
imageC1C2C3C3C3C6C6C6C9oHe3C2xC9oHe3
kernelC2xC9oHe3C9oHe3C3xC18C2xHe3C2x3- 1+2C3xC9He33- 1+2C2C1
# reps118216821666

Matrix representation of C2xC9oHe3 in GL3(F19) generated by

1800
0180
0018
,
400
040
004
,
1112
0110
007
,
1100
0110
0011
,
1112
007
101818
G:=sub<GL(3,GF(19))| [18,0,0,0,18,0,0,0,18],[4,0,0,0,4,0,0,0,4],[1,0,0,1,11,0,12,0,7],[11,0,0,0,11,0,0,0,11],[1,0,10,1,0,18,12,7,18] >;

C2xC9oHe3 in GAP, Magma, Sage, TeX

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

G:=Group("C2xC9oHe3");
// GroupNames label

G:=SmallGroup(162,50);
// by ID

G=gap.SmallGroup(162,50);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,457,78]);
// Polycyclic

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

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