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G = C3×S3×F5order 360 = 23·32·5

Direct product of C3, S3 and F5

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

Aliases: C3×S3×F5, D15⋊C12, C3⋊F5⋊C6, C5⋊(S3×C12), (C3×F5)⋊C6, (C5×S3)⋊C12, C15⋊(C2×C12), (S3×D5).C6, C31(C6×F5), C154(C4×S3), (S3×C15)⋊2C4, C325(C2×F5), (C3×D15)⋊1C4, D5.1(S3×C6), (C3×D5).5D6, (C32×F5)⋊1C2, (C32×D5).1C22, (C3×C3⋊F5)⋊1C2, (C3×C15)⋊3(C2×C4), (C3×D5).(C2×C6), (C3×S3×D5).2C2, SmallGroup(360,126)

Series: Derived Chief Lower central Upper central

C1C15 — C3×S3×F5
C1C5C15C3×D5C32×D5C32×F5 — C3×S3×F5
C15 — C3×S3×F5
C1C3

Generators and relations for C3×S3×F5
 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, C2×C4, C32, D5, D5, C10, Dic3, C12, D6, C2×C6, C15, C15, C3×S3, C3×S3, C3×C6, F5, F5, D10, C4×S3, C2×C12, C5×S3, C3×D5, C3×D5, D15, C30, C3×Dic3, C3×C12, S3×C6, C2×F5, C3×C15, C3×F5, C3×F5, C3⋊F5, S3×D5, C6×D5, S3×C12, C32×D5, S3×C15, C3×D15, S3×F5, C6×F5, C32×F5, C3×C3⋊F5, C3×S3×D5, C3×S3×F5
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C12, D6, C2×C6, C3×S3, F5, C4×S3, C2×C12, S3×C6, C2×F5, C3×F5, S3×C12, S3×F5, C6×F5, C3×S3×F5

Permutation representations of C3×S3×F5
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+++++++++
imageC1C2C2C2C3C4C4C6C6C6C12C12S3D6C3×S3C4×S3S3×C6S3×C12F5C2×F5C3×F5C6×F5S3×F5C3×S3×F5
kernelC3×S3×F5C32×F5C3×C3⋊F5C3×S3×D5S3×F5S3×C15C3×D15C3×F5C3⋊F5S3×D5C5×S3D15C3×F5C3×D5F5C15D5C5C3×S3C32S3C3C3C1
# reps111122222244112224112212

Matrix representation of C3×S3×F5 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] >;

C3×S3×F5 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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