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G = C3xS3xF5order 360 = 23·32·5

Direct product of C3, S3 and F5

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C3xS3xF5, D15:C12, C3:F5:C6, C5:(S3xC12), (C3xF5):C6, (C5xS3):C12, C15:(C2xC12), (S3xD5).C6, C3:1(C6xF5), C15:4(C4xS3), (S3xC15):2C4, C32:5(C2xF5), (C3xD15):1C4, D5.1(S3xC6), (C3xD5).5D6, (C32xF5):1C2, (C32xD5).1C22, (C3xC3:F5):1C2, (C3xC15):3(C2xC4), (C3xD5).(C2xC6), (C3xS3xD5).2C2, SmallGroup(360,126)

Series: Derived Chief Lower central Upper central

C1C15 — C3xS3xF5
C1C5C15C3xD5C32xD5C32xF5 — C3xS3xF5
C15 — C3xS3xF5
C1C3

Generators and relations for C3xS3xF5
 G = < a,b,c,d,e | a3=b3=c2=d5=e4=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d3 >

Subgroups: 292 in 70 conjugacy classes, 28 normal (all characteristic)
C1, C2, C3, C3, C4, C22, C5, S3, S3, C6, C2xC4, C32, D5, D5, C10, Dic3, C12, D6, C2xC6, C15, C15, C3xS3, C3xS3, C3xC6, F5, F5, D10, C4xS3, C2xC12, C5xS3, C3xD5, C3xD5, D15, C30, C3xDic3, C3xC12, S3xC6, C2xF5, C3xC15, C3xF5, C3xF5, C3:F5, S3xD5, C6xD5, S3xC12, C32xD5, S3xC15, C3xD15, S3xF5, C6xF5, C32xF5, C3xC3:F5, C3xS3xD5, C3xS3xF5
Quotients: C1, C2, C3, C4, C22, S3, C6, C2xC4, C12, D6, C2xC6, C3xS3, F5, C4xS3, C2xC12, S3xC6, C2xF5, C3xF5, S3xC12, S3xF5, C6xF5, C3xS3xF5

Permutation representations of C3xS3xF5
On 30 points - transitive group 30T91
Generators in S30
(1 6 11)(2 7 12)(3 8 13)(4 9 14)(5 10 15)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)
(1 6 11)(2 7 12)(3 8 13)(4 9 14)(5 10 15)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)
(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)
(2 3 5 4)(7 8 10 9)(12 13 15 14)(17 18 20 19)(22 23 25 24)(27 28 30 29)

G:=sub<Sym(30)| (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25), (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30), (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), (2,3,5,4)(7,8,10,9)(12,13,15,14)(17,18,20,19)(22,23,25,24)(27,28,30,29)>;

G:=Group( (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25), (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30), (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), (2,3,5,4)(7,8,10,9)(12,13,15,14)(17,18,20,19)(22,23,25,24)(27,28,30,29) );

G=PermutationGroup([[(1,6,11),(2,7,12),(3,8,13),(4,9,14),(5,10,15),(16,21,26),(17,22,27),(18,23,28),(19,24,29),(20,25,30)], [(1,6,11),(2,7,12),(3,8,13),(4,9,14),(5,10,15),(16,26,21),(17,27,22),(18,28,23),(19,29,24),(20,30,25)], [(1,16),(2,17),(3,18),(4,19),(5,20),(6,21),(7,22),(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,29),(15,30)], [(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)], [(2,3,5,4),(7,8,10,9),(12,13,15,14),(17,18,20,19),(22,23,25,24),(27,28,30,29)]])

G:=TransitiveGroup(30,91);

45 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D 5 6A6B6C6D6E6F6G6H6I 10 12A12B12C12D12E···12J12K12L12M12N15A15B15C15D15E30A30B
order12223333344445666666666101212121212···121212121215151515153030
size135151122255151543355101010151512555510···1015151515448881212

45 irreducible representations

dim111111111111222222444488
type+++++++++
imageC1C2C2C2C3C4C4C6C6C6C12C12S3D6C3xS3C4xS3S3xC6S3xC12F5C2xF5C3xF5C6xF5S3xF5C3xS3xF5
kernelC3xS3xF5C32xF5C3xC3:F5C3xS3xD5S3xF5S3xC15C3xD15C3xF5C3:F5S3xD5C5xS3D15C3xF5C3xD5F5C15D5C5C3xS3C32S3C3C3C1
# reps111122222244112224112212

Matrix representation of C3xS3xF5 in GL6(F61)

4700000
0470000
001000
000100
000010
000001
,
4700000
19130000
001000
000100
000010
000001
,
6050000
010000
001000
000100
000010
000001
,
100000
010000
0000060
0010060
0001060
0000160
,
100000
010000
000010
001000
000001
000100

G:=sub<GL(6,GF(61))| [47,0,0,0,0,0,0,47,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[47,19,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,5,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,60,60,60,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;

C3xS3xF5 in GAP, Magma, Sage, TeX

C_3\times S_3\times F_5
% in TeX

G:=Group("C3xS3xF5");
// GroupNames label

G:=SmallGroup(360,126);
// by ID

G=gap.SmallGroup(360,126);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-5,72,730,5189,887]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^5=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^3>;
// generators/relations

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