Copied to
clipboard

G = D5×C3⋊S3order 180 = 22·32·5

Direct product of D5 and C3⋊S3

Aliases: D5×C3⋊S3, C151D6, C324D10, (C3×D5)⋊S3, C32(S3×D5), C3⋊D151C2, (C3×C15)⋊2C22, (C32×D5)⋊2C2, C51(C2×C3⋊S3), (C5×C3⋊S3)⋊1C2, SmallGroup(180,27)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C15 — D5×C3⋊S3
 Chief series C1 — C5 — C15 — C3×C15 — C32×D5 — D5×C3⋊S3
 Lower central C3×C15 — D5×C3⋊S3
 Upper central C1

Generators and relations for D5×C3⋊S3
G = < a,b,c,d,e | a5=b2=c3=d3=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 356 in 60 conjugacy classes, 22 normal (10 characteristic)
C1, C2, C3, C22, C5, S3, C6, C32, D5, D5, C10, D6, C15, C3⋊S3, C3⋊S3, C3×C6, D10, C5×S3, C3×D5, D15, C2×C3⋊S3, C3×C15, S3×D5, C32×D5, C5×C3⋊S3, C3⋊D15, D5×C3⋊S3
Quotients: C1, C2, C22, S3, D5, D6, C3⋊S3, D10, C2×C3⋊S3, S3×D5, D5×C3⋊S3

Character table of D5×C3⋊S3

 class 1 2A 2B 2C 3A 3B 3C 3D 5A 5B 6A 6B 6C 6D 10A 10B 15A 15B 15C 15D 15E 15F 15G 15H size 1 5 9 45 2 2 2 2 2 2 10 10 10 10 18 18 4 4 4 4 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 2 -2 0 0 -1 -1 2 -1 2 2 1 1 -2 1 0 0 -1 2 2 -1 -1 -1 -1 -1 orthogonal lifted from D6 ρ6 2 2 0 0 -1 2 -1 -1 2 2 -1 2 -1 -1 0 0 -1 -1 -1 2 -1 -1 2 -1 orthogonal lifted from S3 ρ7 2 -2 0 0 2 -1 -1 -1 2 2 1 1 1 -2 0 0 -1 -1 -1 -1 -1 2 -1 2 orthogonal lifted from D6 ρ8 2 -2 0 0 -1 -1 -1 2 2 2 -2 1 1 1 0 0 2 -1 -1 -1 2 -1 -1 -1 orthogonal lifted from D6 ρ9 2 2 0 0 -1 -1 2 -1 2 2 -1 -1 2 -1 0 0 -1 2 2 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 0 0 2 -1 -1 -1 2 2 -1 -1 -1 2 0 0 -1 -1 -1 -1 -1 2 -1 2 orthogonal lifted from S3 ρ11 2 2 0 0 -1 -1 -1 2 2 2 2 -1 -1 -1 0 0 2 -1 -1 -1 2 -1 -1 -1 orthogonal lifted from S3 ρ12 2 -2 0 0 -1 2 -1 -1 2 2 1 -2 1 1 0 0 -1 -1 -1 2 -1 -1 2 -1 orthogonal lifted from D6 ρ13 2 0 2 0 2 2 2 2 -1-√5/2 -1+√5/2 0 0 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 ρ14 2 0 -2 0 2 2 2 2 -1+√5/2 -1-√5/2 0 0 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 0 -2 0 2 2 2 2 -1-√5/2 -1+√5/2 0 0 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 ρ16 2 0 2 0 2 2 2 2 -1+√5/2 -1-√5/2 0 0 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 4 0 0 0 -2 -2 4 -2 -1-√5 -1+√5 0 0 0 0 0 0 1-√5/2 -1-√5 -1+√5 1+√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 orthogonal lifted from S3×D5 ρ18 4 0 0 0 -2 4 -2 -2 -1+√5 -1-√5 0 0 0 0 0 0 1+√5/2 1-√5/2 1+√5/2 -1+√5 1-√5/2 1-√5/2 -1-√5 1+√5/2 orthogonal lifted from S3×D5 ρ19 4 0 0 0 -2 4 -2 -2 -1-√5 -1+√5 0 0 0 0 0 0 1-√5/2 1+√5/2 1-√5/2 -1-√5 1+√5/2 1+√5/2 -1+√5 1-√5/2 orthogonal lifted from S3×D5 ρ20 4 0 0 0 -2 -2 4 -2 -1+√5 -1-√5 0 0 0 0 0 0 1+√5/2 -1+√5 -1-√5 1-√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 orthogonal lifted from S3×D5 ρ21 4 0 0 0 -2 -2 -2 4 -1+√5 -1-√5 0 0 0 0 0 0 -1-√5 1-√5/2 1+√5/2 1-√5/2 -1+√5 1-√5/2 1+√5/2 1+√5/2 orthogonal lifted from S3×D5 ρ22 4 0 0 0 4 -2 -2 -2 -1-√5 -1+√5 0 0 0 0 0 0 1-√5/2 1+√5/2 1-√5/2 1+√5/2 1+√5/2 -1-√5 1-√5/2 -1+√5 orthogonal lifted from S3×D5 ρ23 4 0 0 0 -2 -2 -2 4 -1-√5 -1+√5 0 0 0 0 0 0 -1+√5 1+√5/2 1-√5/2 1+√5/2 -1-√5 1+√5/2 1-√5/2 1-√5/2 orthogonal lifted from S3×D5 ρ24 4 0 0 0 4 -2 -2 -2 -1+√5 -1-√5 0 0 0 0 0 0 1+√5/2 1-√5/2 1+√5/2 1-√5/2 1-√5/2 -1+√5 1+√5/2 -1-√5 orthogonal lifted from S3×D5

Smallest permutation representation of D5×C3⋊S3
On 45 points
Generators in S45
(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)(31 32 33 34 35)(36 37 38 39 40)(41 42 43 44 45)
(1 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)(32 35)(33 34)(37 40)(38 39)(42 45)(43 44)
(1 14 9)(2 15 10)(3 11 6)(4 12 7)(5 13 8)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)(31 41 36)(32 42 37)(33 43 38)(34 44 39)(35 45 40)
(1 19 34)(2 20 35)(3 16 31)(4 17 32)(5 18 33)(6 21 36)(7 22 37)(8 23 38)(9 24 39)(10 25 40)(11 26 41)(12 27 42)(13 28 43)(14 29 44)(15 30 45)
(6 11)(7 12)(8 13)(9 14)(10 15)(16 31)(17 32)(18 33)(19 34)(20 35)(21 41)(22 42)(23 43)(24 44)(25 45)(26 36)(27 37)(28 38)(29 39)(30 40)

G:=sub<Sym(45)| (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)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)(37,40)(38,39)(42,45)(43,44), (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25)(31,41,36)(32,42,37)(33,43,38)(34,44,39)(35,45,40), (1,19,34)(2,20,35)(3,16,31)(4,17,32)(5,18,33)(6,21,36)(7,22,37)(8,23,38)(9,24,39)(10,25,40)(11,26,41)(12,27,42)(13,28,43)(14,29,44)(15,30,45), (6,11)(7,12)(8,13)(9,14)(10,15)(16,31)(17,32)(18,33)(19,34)(20,35)(21,41)(22,42)(23,43)(24,44)(25,45)(26,36)(27,37)(28,38)(29,39)(30,40)>;

G:=Group( (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)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)(37,40)(38,39)(42,45)(43,44), (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25)(31,41,36)(32,42,37)(33,43,38)(34,44,39)(35,45,40), (1,19,34)(2,20,35)(3,16,31)(4,17,32)(5,18,33)(6,21,36)(7,22,37)(8,23,38)(9,24,39)(10,25,40)(11,26,41)(12,27,42)(13,28,43)(14,29,44)(15,30,45), (6,11)(7,12)(8,13)(9,14)(10,15)(16,31)(17,32)(18,33)(19,34)(20,35)(21,41)(22,42)(23,43)(24,44)(25,45)(26,36)(27,37)(28,38)(29,39)(30,40) );

G=PermutationGroup([[(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),(31,32,33,34,35),(36,37,38,39,40),(41,42,43,44,45)], [(1,5),(2,4),(7,10),(8,9),(12,15),(13,14),(17,20),(18,19),(22,25),(23,24),(27,30),(28,29),(32,35),(33,34),(37,40),(38,39),(42,45),(43,44)], [(1,14,9),(2,15,10),(3,11,6),(4,12,7),(5,13,8),(16,26,21),(17,27,22),(18,28,23),(19,29,24),(20,30,25),(31,41,36),(32,42,37),(33,43,38),(34,44,39),(35,45,40)], [(1,19,34),(2,20,35),(3,16,31),(4,17,32),(5,18,33),(6,21,36),(7,22,37),(8,23,38),(9,24,39),(10,25,40),(11,26,41),(12,27,42),(13,28,43),(14,29,44),(15,30,45)], [(6,11),(7,12),(8,13),(9,14),(10,15),(16,31),(17,32),(18,33),(19,34),(20,35),(21,41),(22,42),(23,43),(24,44),(25,45),(26,36),(27,37),(28,38),(29,39),(30,40)]])

D5×C3⋊S3 is a maximal subgroup of   C3⋊F5⋊S3  C32⋊F5⋊C2  S32×D5
D5×C3⋊S3 is a maximal quotient of   C30.D6  C30.12D6  C327D20  C15⋊D12  C15⋊Dic6

Matrix representation of D5×C3⋊S3 in GL6(𝔽31)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 19 1 0 0 0 0 11 30 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 30 30 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 29 21 0 0 0 0 22 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 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 30 0 0 0 0 1 30
,
 30 21 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 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(31))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,19,11,0,0,0,0,1,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,0,0,0,0,0,30,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[29,22,0,0,0,0,21,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,30,30],[30,0,0,0,0,0,21,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

D5×C3⋊S3 in GAP, Magma, Sage, TeX

D_5\times C_3\rtimes S_3
% in TeX

G:=Group("D5xC3:S3");
// GroupNames label

G:=SmallGroup(180,27);
// by ID

G=gap.SmallGroup(180,27);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-5,67,248,3604]);
// Polycyclic

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

Export

׿
×
𝔽