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G = C3xD52order 300 = 22·3·52

Direct product of C3, D5 and D5

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

Aliases: C3xD52, C15:4D10, (C5xD5):C6, C5:D5:2C6, C5:1(C6xD5), C52:2(C2xC6), (D5xC15):2C2, (C5xC15):5C22, (C3xC5:D5):3C2, SmallGroup(300,36)

Series: Derived Chief Lower central Upper central

C1C52 — C3xD52
C1C5C52C5xC15D5xC15 — C3xD52
C52 — C3xD52
C1C3

Generators and relations for C3xD52
 G = < a,b,c,d,e | a3=b5=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 248 in 48 conjugacy classes, 20 normal (8 characteristic)
Quotients: C1, C2, C3, C22, C6, D5, C2xC6, D10, C3xD5, C6xD5, D52, C3xD52
5C2
5C2
25C2
2C5
2C5
25C22
5C6
5C6
25C6
5C10
5C10
5D5
5D5
10D5
10D5
2C15
2C15
25C2xC6
5D10
5D10
5C30
5C3xD5
5C3xD5
5C30
10C3xD5
10C3xD5
5C6xD5
5C6xD5

Permutation representations of C3xD52
On 30 points - transitive group 30T74
Generators in S30
(1 11 6)(2 12 7)(3 13 8)(4 14 9)(5 15 10)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)
(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)
(1 18)(2 17)(3 16)(4 20)(5 19)(6 23)(7 22)(8 21)(9 25)(10 24)(11 28)(12 27)(13 26)(14 30)(15 29)
(1 5 4 3 2)(6 10 9 8 7)(11 15 14 13 12)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)
(1 18)(2 19)(3 20)(4 16)(5 17)(6 23)(7 24)(8 25)(9 21)(10 22)(11 28)(12 29)(13 30)(14 26)(15 27)

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

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

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

G:=TransitiveGroup(30,74);

48 conjugacy classes

class 1 2A2B2C3A3B5A5B5C5D5E5F5G5H6A6B6C6D6E6F10A10B10C10D15A···15H15I···15P30A···30H
order122233555555556666661010101015···1515···1530···30
size15525112222444455552525101010102···24···410···10

48 irreducible representations

dim111111222244
type++++++
imageC1C2C2C3C6C6D5D10C3xD5C6xD5D52C3xD52
kernelC3xD52D5xC15C3xC5:D5D52C5xD5C5:D5C3xD5C15D5C5C3C1
# reps121242448848

Matrix representation of C3xD52 in GL4(F31) generated by

5000
0500
0010
0001
,
1000
0100
00030
00112
,
30000
03000
003019
0001
,
03000
11800
0010
0001
,
301300
0100
00300
00030
G:=sub<GL(4,GF(31))| [5,0,0,0,0,5,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,30,12],[30,0,0,0,0,30,0,0,0,0,30,0,0,0,19,1],[0,1,0,0,30,18,0,0,0,0,1,0,0,0,0,1],[30,0,0,0,13,1,0,0,0,0,30,0,0,0,0,30] >;

C3xD52 in GAP, Magma, Sage, TeX

C_3\times D_5^2
% in TeX

G:=Group("C3xD5^2");
// GroupNames label

G:=SmallGroup(300,36);
// by ID

G=gap.SmallGroup(300,36);
# by ID

G:=PCGroup([5,-2,-2,-3,-5,-5,488,6004]);
// Polycyclic

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

Export

Subgroup lattice of C3xD52 in TeX

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