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## G = C2×S3×D5order 120 = 23·3·5

### Direct product of C2, S3 and D5

Aliases: C2×S3×D5, C15⋊C23, C61D10, C101D6, C30⋊C22, D305C2, D15⋊C22, (C6×D5)⋊3C2, (C5×S3)⋊C22, C51(C22×S3), (C3×D5)⋊C22, (S3×C10)⋊3C2, C31(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 [×6], C3, C22 [×7], C5, S3 [×2], S3 [×2], C6, C6 [×2], C23, D5 [×2], D5 [×2], C10, C10 [×2], D6, D6 [×5], C2×C6, C15, D10, D10 [×5], C2×C10, C22×S3, C5×S3 [×2], C3×D5 [×2], D15 [×2], C30, C22×D5, S3×D5 [×4], C6×D5, S3×C10, D30, C2×S3×D5
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D5, D6 [×3], D10 [×3], C22×S3, C22×D5, S3×D5, C2×S3×D5

Character table of C2×S3×D5

 class 1 2A 2B 2C 2D 2E 2F 2G 3 5A 5B 6A 6B 6C 10A 10B 10C 10D 10E 10F 15A 15B 30A 30B size 1 1 3 3 5 5 15 15 2 2 2 2 10 10 2 2 6 6 6 6 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 -1 -1 1 -1 1 1 -1 1 1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 -1 linear of order 2 ρ4 1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 linear of order 2 ρ5 1 -1 1 -1 -1 1 -1 1 1 1 1 -1 1 -1 -1 -1 1 1 -1 -1 1 1 -1 -1 linear of order 2 ρ6 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 -1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 linear of order 2 ρ7 1 1 -1 -1 1 1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ8 1 1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ9 2 2 0 0 -2 -2 0 0 -1 2 2 -1 1 1 2 2 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from D6 ρ10 2 -2 0 0 2 -2 0 0 -1 2 2 1 1 -1 -2 -2 0 0 0 0 -1 -1 1 1 orthogonal lifted from D6 ρ11 2 2 0 0 2 2 0 0 -1 2 2 -1 -1 -1 2 2 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 -2 0 0 -2 2 0 0 -1 2 2 1 -1 1 -2 -2 0 0 0 0 -1 -1 1 1 orthogonal lifted from D6 ρ13 2 -2 2 -2 0 0 0 0 2 -1-√5/2 -1+√5/2 -2 0 0 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 D10 ρ14 2 -2 2 -2 0 0 0 0 2 -1+√5/2 -1-√5/2 -2 0 0 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 D10 ρ15 2 2 2 2 0 0 0 0 2 -1-√5/2 -1+√5/2 2 0 0 -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 ρ16 2 2 2 2 0 0 0 0 2 -1+√5/2 -1-√5/2 2 0 0 -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 ρ17 2 2 -2 -2 0 0 0 0 2 -1+√5/2 -1-√5/2 2 0 0 -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 D10 ρ18 2 -2 -2 2 0 0 0 0 2 -1-√5/2 -1+√5/2 -2 0 0 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 D10 ρ19 2 -2 -2 2 0 0 0 0 2 -1+√5/2 -1-√5/2 -2 0 0 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 D10 ρ20 2 2 -2 -2 0 0 0 0 2 -1-√5/2 -1+√5/2 2 0 0 -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 D10 ρ21 4 4 0 0 0 0 0 0 -2 -1+√5 -1-√5 -2 0 0 -1-√5 -1+√5 0 0 0 0 1-√5/2 1+√5/2 1+√5/2 1-√5/2 orthogonal lifted from S3×D5 ρ22 4 -4 0 0 0 0 0 0 -2 -1+√5 -1-√5 2 0 0 1+√5 1-√5 0 0 0 0 1-√5/2 1+√5/2 -1-√5/2 -1+√5/2 orthogonal faithful ρ23 4 4 0 0 0 0 0 0 -2 -1-√5 -1+√5 -2 0 0 -1+√5 -1-√5 0 0 0 0 1+√5/2 1-√5/2 1-√5/2 1+√5/2 orthogonal lifted from S3×D5 ρ24 4 -4 0 0 0 0 0 0 -2 -1-√5 -1+√5 2 0 0 1-√5 1+√5 0 0 0 0 1+√5/2 1-√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
D205S3  D20⋊S3  D12⋊D5  D60⋊C2  D15⋊Q8  D6.D10  D125D5  C12.28D10  C20⋊D6  Dic5.D6  C30.C23  Dic3.D10  D10⋊D6

Matrix representation of C2×S3×D5 in GL4(𝔽31) generated by

 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 10 0 0 9 29
,
 1 0 0 0 0 1 0 0 0 0 30 21 0 0 0 1
,
 19 1 0 0 11 30 0 0 0 0 1 0 0 0 0 1
,
 30 30 0 0 0 1 0 0 0 0 1 0 0 0 0 1
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

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