metabelian, supersoluble, monomial
Aliases: He3⋊1D5, C32⋊1D15, C3⋊D15⋊C3, C5⋊(C32⋊C6), C32⋊(C3×D5), (C3×C15)⋊1S3, (C3×C15)⋊1C6, (C5×He3)⋊1C2, C15.2(C3×S3), C3.2(C3×D15), SmallGroup(270,14)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C15 — He3⋊D5 |
Generators and relations for He3⋊D5
G = < a,b,c,d,e | a3=b3=c3=d5=e2=1, ab=ba, cac-1=ab-1, ad=da, eae=a-1, bc=cb, bd=db, ebe=b-1, cd=dc, ce=ec, ede=d-1 >
Smallest permutation representation of He3⋊D5
►On 45 points
Generators in S45
(1 34 19)(2 35 20)(3 31 16)(4 32 17)(5 33 18)(6 36 21)(7 37 22)(8 38 23)(9 39 24)(10 40 25)(11 41 26)(12 42 27)(13 43 28)(14 44 29)(15 45 30)
(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)
(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)(31 41 36)(32 42 37)(33 43 38)(34 44 39)(35 45 40)
(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)(6 11)(7 15)(8 14)(9 13)(10 12)(16 31)(17 35)(18 34)(19 33)(20 32)(21 41)(22 45)(23 44)(24 43)(25 42)(26 36)(27 40)(28 39)(29 38)(30 37)
G:=sub<Sym(45)| (1,34,19)(2,35,20)(3,31,16)(4,32,17)(5,33,18)(6,36,21)(7,37,22)(8,38,23)(9,39,24)(10,40,25)(11,41,26)(12,42,27)(13,43,28)(14,44,29)(15,45,30), (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), (16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30)(31,41,36)(32,42,37)(33,43,38)(34,44,39)(35,45,40), (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)(6,11)(7,15)(8,14)(9,13)(10,12)(16,31)(17,35)(18,34)(19,33)(20,32)(21,41)(22,45)(23,44)(24,43)(25,42)(26,36)(27,40)(28,39)(29,38)(30,37)>;
G:=Group( (1,34,19)(2,35,20)(3,31,16)(4,32,17)(5,33,18)(6,36,21)(7,37,22)(8,38,23)(9,39,24)(10,40,25)(11,41,26)(12,42,27)(13,43,28)(14,44,29)(15,45,30), (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), (16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30)(31,41,36)(32,42,37)(33,43,38)(34,44,39)(35,45,40), (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)(6,11)(7,15)(8,14)(9,13)(10,12)(16,31)(17,35)(18,34)(19,33)(20,32)(21,41)(22,45)(23,44)(24,43)(25,42)(26,36)(27,40)(28,39)(29,38)(30,37) );
G=PermutationGroup([[(1,34,19),(2,35,20),(3,31,16),(4,32,17),(5,33,18),(6,36,21),(7,37,22),(8,38,23),(9,39,24),(10,40,25),(11,41,26),(12,42,27),(13,43,28),(14,44,29),(15,45,30)], [(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)], [(16,21,26),(17,22,27),(18,23,28),(19,24,29),(20,25,30),(31,41,36),(32,42,37),(33,43,38),(34,44,39),(35,45,40)], [(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),(6,11),(7,15),(8,14),(9,13),(10,12),(16,31),(17,35),(18,34),(19,33),(20,32),(21,41),(22,45),(23,44),(24,43),(25,42),(26,36),(27,40),(28,39),(29,38),(30,37)]])
32 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 5A | 5B | 6A | 6B | 15A | 15B | 15C | 15D | 15E | ··· | 15T |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 5 | 5 | 6 | 6 | 15 | 15 | 15 | 15 | 15 | ··· | 15 |
| size | 1 | 45 | 2 | 3 | 3 | 6 | 6 | 6 | 2 | 2 | 45 | 45 | 2 | 2 | 2 | 2 | 6 | ··· | 6 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 |
| type | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C3 | C6 | S3 | D5 | C3×S3 | C3×D5 | D15 | C3×D15 | C32⋊C6 | He3⋊D5 |
| kernel | He3⋊D5 | C5×He3 | C3⋊D15 | C3×C15 | C3×C15 | He3 | C15 | C32 | C32 | C3 | C5 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 4 | 4 | 8 | 1 | 4 |
Matrix representation of He3⋊D5 ►in GL6(𝔽31)
| 14 | 3 | 0 | 25 | 0 | 0 |
| 2 | 16 | 6 | 6 | 0 | 0 |
| 0 | 0 | 12 | 28 | 1 | 0 |
| 0 | 0 | 3 | 20 | 0 | 1 |
| 0 | 0 | 20 | 3 | 0 | 0 |
| 0 | 0 | 28 | 12 | 0 | 0 |
| 19 | 5 | 0 | 0 | 0 | 0 |
| 23 | 11 | 0 | 0 | 0 | 0 |
| 16 | 0 | 16 | 5 | 0 | 0 |
| 15 | 15 | 26 | 14 | 0 | 0 |
| 15 | 0 | 0 | 0 | 16 | 5 |
| 16 | 16 | 0 | 0 | 26 | 14 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 5 | 23 | 14 | 26 | 0 | 0 |
| 30 | 29 | 5 | 16 | 0 | 0 |
| 1 | 30 | 0 | 0 | 16 | 5 |
| 26 | 18 | 0 | 0 | 26 | 14 |
| 13 | 1 | 0 | 0 | 0 | 0 |
| 17 | 30 | 0 | 0 | 0 | 0 |
| 28 | 0 | 0 | 1 | 0 | 0 |
| 3 | 3 | 30 | 12 | 0 | 0 |
| 3 | 0 | 0 | 0 | 0 | 1 |
| 28 | 28 | 0 | 0 | 30 | 12 |
| 30 | 30 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
G:=sub<GL(6,GF(31))| [14,2,0,0,0,0,3,16,0,0,0,0,0,6,12,3,20,28,25,6,28,20,3,12,0,0,1,0,0,0,0,0,0,1,0,0],[19,23,16,15,15,16,5,11,0,15,0,16,0,0,16,26,0,0,0,0,5,14,0,0,0,0,0,0,16,26,0,0,0,0,5,14],[1,0,5,30,1,26,0,1,23,29,30,18,0,0,14,5,0,0,0,0,26,16,0,0,0,0,0,0,16,26,0,0,0,0,5,14],[13,17,28,3,3,28,1,30,0,3,0,28,0,0,0,30,0,0,0,0,1,12,0,0,0,0,0,0,0,30,0,0,0,0,1,12],[30,0,0,0,0,0,30,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0] >;
He3⋊D5 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes D_5 % in TeX
G:=Group("He3:D5"); // GroupNames label
G:=SmallGroup(270,14);
// by ID
G=gap.SmallGroup(270,14);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-5,182,187,723,5404]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^5=e^2=1,a*b=b*a,c*a*c^-1=a*b^-1,a*d=d*a,e*a*e=a^-1,b*c=c*b,b*d=d*b,e*b*e=b^-1,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations
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