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G = C2×S3×D5  order 120 = 23·3·5

Direct product of C2, S3 and D5

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

Aliases: C2×S3×D5, C15⋊C23, C6⋊1D10, C10⋊1D6, C30⋊C22, D30⋊5C2, D15⋊C22, (C6×D5)⋊3C2, (C5×S3)⋊C22, C5⋊1(C22×S3), (C3×D5)⋊C22, (S3×C10)⋊3C2, C3⋊1(C22×D5), SmallGroup(120,42)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C15 — C2×S3×D5
Chief series
C1 — C5 — C15 — C3×D5 — S3×D5 — C2×S3×D5
Lower central
C15 — C2×S3×D5
Upper central
C1 — C2

Generators and relations for C2×S3×D5
 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, C2×C6, C15, D10, D10, C2×C10, C22×S3, C5×S3, C3×D5, D15, C30, C22×D5, S3×D5, C6×D5, S3×C10, D30, C2×S3×D5
Quotients: C1, C2, C22, S3, C23, D5, D6, D10, C22×S3, C22×D5, S3×D5, C2×S3×D5

Character table of C2×S3×D5

 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 S3×D5
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 S3×D5
ρ244-4000000-2-1-√5-1+√52001-√51+√500001+√5/21-√5/2-1+√5/2-1-√5/2    orthogonal faithful

Permutation representations of C2×S3×D5
►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);
 

C2×S3×D5 is a maximal subgroup of
 D6⋊F5  C20⋊D6  D10⋊D6
C2×S3×D5 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 C2×S3×D5 ►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] >;
 

C2×S3×D5 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 C2×S3×D5 in TeX

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