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G = C2xS3xD5  order 120 = 23·3·5

Direct product of C2, S3 and D5

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C2xS3xD5, C15:C23, C6:1D10, C10:1D6, C30:C22, D30:5C2, D15:C22, (C6xD5):3C2, (C5xS3):C22, C5:1(C22xS3), (C3xD5):C22, (S3xC10):3C2, C3:1(C22xD5), SmallGroup(120,42)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C15 — C2xS3xD5
Chief series
C1 — C5 — C15 — C3xD5 — S3xD5 — C2xS3xD5
Lower central
C15 — C2xS3xD5
Upper central
C1 — C2

Generators and relations for C2xS3xD5
 G = < a,b,c,d,e | a2=b3=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: 260 in 64 conjugacy classes, 28 normal (14 characteristic)
C1, C2, C2, C3, C22, C5, S3, S3, C6, C6, C23, D5, D5, C10, C10, D6, D6, C2xC6, C15, D10, D10, C2xC10, C22xS3, C5xS3, C3xD5, D15, C30, C22xD5, S3xD5, C6xD5, S3xC10, D30, C2xS3xD5
Quotients: C1, C2, C22, S3, C23, D5, D6, D10, C22xS3, C22xD5, S3xD5, C2xS3xD5

Character table of C2xS3xD5

 class 12A2B2C2D2E2F2G35A5B6A6B6C10A10B10C10D10E10F15A15B30A30B
 size 1133551515222210102266664444
ρ1111111111111111111111111    trivial
ρ21111-1-1-1-11111-1-11111111111    linear of order 2
ρ31-1-11-111-1111-11-1-1-1-1-11111-1-1    linear of order 2
ρ41-1-111-1-11111-1-11-1-1-1-11111-1-1    linear of order 2
ρ51-11-1-11-11111-11-1-1-111-1-111-1-1    linear of order 2
ρ61-11-11-11-1111-1-11-1-111-1-111-1-1    linear of order 2
ρ711-1-111-1-111111111-1-1-1-11111    linear of order 2
ρ811-1-1-1-1111111-1-111-1-1-1-11111    linear of order 2
ρ92200-2-200-122-111220000-1-1-1-1    orthogonal lifted from D6
ρ102-2002-200-12211-1-2-20000-1-111    orthogonal lifted from D6
ρ1122002200-122-1-1-1220000-1-1-1-1    orthogonal lifted from S3
ρ122-200-2200-1221-11-2-20000-1-111    orthogonal lifted from D6
ρ132-22-200002-1-√5/2-1+√5/2-2001-√5/21+√5/2-1-√5/2-1+√5/21-√5/21+√5/2-1-√5/2-1+√5/21-√5/21+√5/2    orthogonal lifted from D10
ρ142-22-200002-1+√5/2-1-√5/2-2001+√5/21-√5/2-1+√5/2-1-√5/21+√5/21-√5/2-1+√5/2-1-√5/21+√5/21-√5/2    orthogonal lifted from D10
ρ15222200002-1-√5/2-1+√5/2200-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2    orthogonal lifted from D5
ρ16222200002-1+√5/2-1-√5/2200-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2    orthogonal lifted from D5
ρ1722-2-200002-1+√5/2-1-√5/2200-1-√5/2-1+√5/21-√5/21+√5/21+√5/21-√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2    orthogonal lifted from D10
ρ182-2-2200002-1-√5/2-1+√5/2-2001-√5/21+√5/21+√5/21-√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/21-√5/21+√5/2    orthogonal lifted from D10
ρ192-2-2200002-1+√5/2-1-√5/2-2001+√5/21-√5/21-√5/21+√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/21+√5/21-√5/2    orthogonal lifted from D10
ρ2022-2-200002-1-√5/2-1+√5/2200-1+√5/2-1-√5/21+√5/21-√5/21-√5/21+√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2    orthogonal lifted from D10
ρ2144000000-2-1+√5-1-√5-200-1-√5-1+√500001-√5/21+√5/21+√5/21-√5/2    orthogonal lifted from S3xD5
ρ224-4000000-2-1+√5-1-√52001+√51-√500001-√5/21+√5/2-1-√5/2-1+√5/2    orthogonal faithful
ρ2344000000-2-1-√5-1+√5-200-1+√5-1-√500001+√5/21-√5/21-√5/21+√5/2    orthogonal lifted from S3xD5
ρ244-4000000-2-1-√5-1+√52001-√51+√500001+√5/21-√5/2-1+√5/2-1-√5/2    orthogonal faithful

Permutation representations of C2xS3xD5
►On 30 points - transitive group 30T21
Generators in S30
(1 19)(2 20)(3 16)(4 17)(5 18)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)

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

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

C2xS3xD5 is a maximal subgroup of
 D6:F5  C20:D6  D10:D6
C2xS3xD5 is a maximal quotient of
 D20:5S3  D20:S3  D12:D5  D60:C2  D15:Q8  D6.D10  D12:5D5  C12.28D10  C20:D6  Dic5.D6  C30.C23  Dic3.D10  D10:D6

Matrix representation of C2xS3xD5 ►in GL4(F31) generated by

30000
03000
00300
00030
,
1000
0100
00110
00929
,
1000
0100
003021
0001
,
19100
113000
0010
0001
,
303000
0100
0010
0001
G:=sub<GL(4,GF(31))| [30,0,0,0,0,30,0,0,0,0,30,0,0,0,0,30],[1,0,0,0,0,1,0,0,0,0,1,9,0,0,10,29],[1,0,0,0,0,1,0,0,0,0,30,0,0,0,21,1],[19,11,0,0,1,30,0,0,0,0,1,0,0,0,0,1],[30,0,0,0,30,1,0,0,0,0,1,0,0,0,0,1] >;
 

C2xS3xD5 in GAP, Magma, Sage, TeX 

C_2\times S_3\times D_5
 
% in TeX
 
G:=Group("C2xS3xD5");
 
// GroupNames label
 
G:=SmallGroup(120,42);
 
// by ID
 
G=gap.SmallGroup(120,42);
 
# by ID
 
G:=PCGroup([5,-2,-2,-2,-3,-5,168,2404]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^3=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

Character table of C2xS3xD5 in TeX

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