direct product, non-abelian, supersoluble, monomial
Aliases: C5×He3⋊C2, He3⋊2C10, (C3×C15)⋊4S3, (C5×He3)⋊5C2, C32⋊2(C5×S3), C15.4(C3⋊S3), C3.2(C5×C3⋊S3), SmallGroup(270,17)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — C5×He3⋊C2 |
Generators and relations for C5×He3⋊C2
G = < a,b,c,d,e | a5=b3=c3=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, ebe=b-1, cd=dc, ce=ec, ede=d-1 >
Smallest permutation representation of C5×He3⋊C2
►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 42 18)(2 43 19)(3 44 20)(4 45 16)(5 41 17)(6 39 12)(7 40 13)(8 36 14)(9 37 15)(10 38 11)(21 33 26)(22 34 27)(23 35 28)(24 31 29)(25 32 30)
(1 8 33)(2 9 34)(3 10 35)(4 6 31)(5 7 32)(11 23 20)(12 24 16)(13 25 17)(14 21 18)(15 22 19)(26 42 36)(27 43 37)(28 44 38)(29 45 39)(30 41 40)
(1 26 21)(2 27 22)(3 28 23)(4 29 24)(5 30 25)(6 45 16)(7 41 17)(8 42 18)(9 43 19)(10 44 20)(11 35 38)(12 31 39)(13 32 40)(14 33 36)(15 34 37)
(11 38)(12 39)(13 40)(14 36)(15 37)(16 45)(17 41)(18 42)(19 43)(20 44)(21 26)(22 27)(23 28)(24 29)(25 30)
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), (1,42,18)(2,43,19)(3,44,20)(4,45,16)(5,41,17)(6,39,12)(7,40,13)(8,36,14)(9,37,15)(10,38,11)(21,33,26)(22,34,27)(23,35,28)(24,31,29)(25,32,30), (1,8,33)(2,9,34)(3,10,35)(4,6,31)(5,7,32)(11,23,20)(12,24,16)(13,25,17)(14,21,18)(15,22,19)(26,42,36)(27,43,37)(28,44,38)(29,45,39)(30,41,40), (1,26,21)(2,27,22)(3,28,23)(4,29,24)(5,30,25)(6,45,16)(7,41,17)(8,42,18)(9,43,19)(10,44,20)(11,35,38)(12,31,39)(13,32,40)(14,33,36)(15,34,37), (11,38)(12,39)(13,40)(14,36)(15,37)(16,45)(17,41)(18,42)(19,43)(20,44)(21,26)(22,27)(23,28)(24,29)(25,30)>;
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), (1,42,18)(2,43,19)(3,44,20)(4,45,16)(5,41,17)(6,39,12)(7,40,13)(8,36,14)(9,37,15)(10,38,11)(21,33,26)(22,34,27)(23,35,28)(24,31,29)(25,32,30), (1,8,33)(2,9,34)(3,10,35)(4,6,31)(5,7,32)(11,23,20)(12,24,16)(13,25,17)(14,21,18)(15,22,19)(26,42,36)(27,43,37)(28,44,38)(29,45,39)(30,41,40), (1,26,21)(2,27,22)(3,28,23)(4,29,24)(5,30,25)(6,45,16)(7,41,17)(8,42,18)(9,43,19)(10,44,20)(11,35,38)(12,31,39)(13,32,40)(14,33,36)(15,34,37), (11,38)(12,39)(13,40)(14,36)(15,37)(16,45)(17,41)(18,42)(19,43)(20,44)(21,26)(22,27)(23,28)(24,29)(25,30) );
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)], [(1,42,18),(2,43,19),(3,44,20),(4,45,16),(5,41,17),(6,39,12),(7,40,13),(8,36,14),(9,37,15),(10,38,11),(21,33,26),(22,34,27),(23,35,28),(24,31,29),(25,32,30)], [(1,8,33),(2,9,34),(3,10,35),(4,6,31),(5,7,32),(11,23,20),(12,24,16),(13,25,17),(14,21,18),(15,22,19),(26,42,36),(27,43,37),(28,44,38),(29,45,39),(30,41,40)], [(1,26,21),(2,27,22),(3,28,23),(4,29,24),(5,30,25),(6,45,16),(7,41,17),(8,42,18),(9,43,19),(10,44,20),(11,35,38),(12,31,39),(13,32,40),(14,33,36),(15,34,37)], [(11,38),(12,39),(13,40),(14,36),(15,37),(16,45),(17,41),(18,42),(19,43),(20,44),(21,26),(22,27),(23,28),(24,29),(25,30)]])
50 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 5A | 5B | 5C | 5D | 6A | 6B | 10A | 10B | 10C | 10D | 15A | ··· | 15H | 15I | ··· | 15X | 30A | ··· | 30H |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 5 | 5 | 5 | 5 | 6 | 6 | 10 | 10 | 10 | 10 | 15 | ··· | 15 | 15 | ··· | 15 | 30 | ··· | 30 |
| size | 1 | 9 | 1 | 1 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 9 | 9 | 9 | 9 | 9 | 9 | 1 | ··· | 1 | 6 | ··· | 6 | 9 | ··· | 9 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 |
| type | + | + | + | |||||
| image | C1 | C2 | C5 | C10 | S3 | C5×S3 | He3⋊C2 | C5×He3⋊C2 |
| kernel | C5×He3⋊C2 | C5×He3 | He3⋊C2 | He3 | C3×C15 | C32 | C5 | C1 |
| # reps | 1 | 1 | 4 | 4 | 4 | 16 | 4 | 16 |
Matrix representation of C5×He3⋊C2 ►in GL3(𝔽31) generated by
| 8 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 5 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 0 | 5 |
| 0 | 5 | 0 |
| 0 | 0 | 1 |
| 25 | 0 | 0 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(31))| [8,0,0,0,8,0,0,0,8],[0,0,1,1,0,0,0,1,0],[5,0,0,0,5,0,0,0,5],[0,0,25,5,0,0,0,1,0],[1,0,0,0,0,1,0,1,0] >;
C5×He3⋊C2 in GAP, Magma, Sage, TeX
C_5\times {\rm He}_3\rtimes C_2 % in TeX
G:=Group("C5xHe3:C2"); // GroupNames label
G:=SmallGroup(270,17);
// by ID
G=gap.SmallGroup(270,17);
# by ID
G:=PCGroup([5,-2,-5,-3,-3,-3,302,1203,253]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^3=c^3=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b*c^-1,e*b*e=b^-1,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations
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