Copied to
clipboard

G = C32×C6order 54 = 2·33

Abelian group of type [3,3,6]

direct product, abelian, monomial, 3-elementary

Aliases: C32×C6, SmallGroup(54,15)

Series: Derived Chief Lower central Upper central

C1 — C32×C6
C1C3C32C33 — C32×C6
C1 — C32×C6
C1 — C32×C6

Generators and relations for C32×C6
 G = < a,b,c | a3=b3=c6=1, ab=ba, ac=ca, bc=cb >

Subgroups: 56, all normal (4 characteristic)
C1, C2, C3 [×13], C6 [×13], C32 [×13], C3×C6 [×13], C33, C32×C6
Quotients: C1, C2, C3 [×13], C6 [×13], C32 [×13], C3×C6 [×13], C33, C32×C6

Smallest permutation representation of C32×C6
Regular action on 54 points
Generators in S54
(1 49 11)(2 50 12)(3 51 7)(4 52 8)(5 53 9)(6 54 10)(13 40 48)(14 41 43)(15 42 44)(16 37 45)(17 38 46)(18 39 47)(19 29 33)(20 30 34)(21 25 35)(22 26 36)(23 27 31)(24 28 32)
(1 20 14)(2 21 15)(3 22 16)(4 23 17)(5 24 18)(6 19 13)(7 36 45)(8 31 46)(9 32 47)(10 33 48)(11 34 43)(12 35 44)(25 42 50)(26 37 51)(27 38 52)(28 39 53)(29 40 54)(30 41 49)
(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)

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

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

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

54 conjugacy classes

class 1  2 3A···3Z6A···6Z
order123···36···6
size111···11···1

54 irreducible representations

dim1111
type++
imageC1C2C3C6
kernelC32×C6C33C3×C6C32
# reps112626

Matrix representation of C32×C6 in GL3(𝔽7) generated by

200
020
004
,
400
010
004
,
200
030
003
G:=sub<GL(3,GF(7))| [2,0,0,0,2,0,0,0,4],[4,0,0,0,1,0,0,0,4],[2,0,0,0,3,0,0,0,3] >;

C32×C6 in GAP, Magma, Sage, TeX

C_3^2\times C_6
% in TeX

G:=Group("C3^2xC6");
// GroupNames label

G:=SmallGroup(54,15);
// by ID

G=gap.SmallGroup(54,15);
# by ID

G:=PCGroup([4,-2,-3,-3,-3]);
// Polycyclic

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

׿
×
𝔽