direct product, metabelian, supersoluble, monomial, A-group
Aliases: D5×C3⋊S3, C15⋊1D6, C32⋊4D10, (C3×D5)⋊S3, C3⋊2(S3×D5), C3⋊D15⋊1C2, (C3×C15)⋊2C22, (C32×D5)⋊2C2, C5⋊1(C2×C3⋊S3), (C5×C3⋊S3)⋊1C2, SmallGroup(180,27)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C15 — D5×C3⋊S3 |
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 |
►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 C32⋊7D20 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
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