direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C5⋊D4, C15⋊8D4, Dic5⋊C6, D10⋊2C6, C6.17D10, C30.17C22, C5⋊2(C3×D4), (C2×C6)⋊1D5, (C2×C30)⋊4C2, (C2×C10)⋊4C6, (C6×D5)⋊5C2, C2.5(C6×D5), C10.5(C2×C6), C22⋊2(C3×D5), (C3×Dic5)⋊4C2, SmallGroup(120,20)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C5⋊D4
G = < a,b,c,d | a3=b5=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Smallest permutation representation of C3×C5⋊D4
►On 60 points
Generators in S60
(1 21 11)(2 22 12)(3 23 13)(4 24 14)(5 25 15)(6 26 16)(7 27 17)(8 28 18)(9 29 19)(10 30 20)(31 51 41)(32 52 42)(33 53 43)(34 54 44)(35 55 45)(36 56 46)(37 57 47)(38 58 48)(39 59 49)(40 60 50)
(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)(46 47 48 49 50)(51 52 53 54 55)(56 57 58 59 60)
(1 31 6 36)(2 35 7 40)(3 34 8 39)(4 33 9 38)(5 32 10 37)(11 41 16 46)(12 45 17 50)(13 44 18 49)(14 43 19 48)(15 42 20 47)(21 51 26 56)(22 55 27 60)(23 54 28 59)(24 53 29 58)(25 52 30 57)
(2 5)(3 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)(31 36)(32 40)(33 39)(34 38)(35 37)(41 46)(42 50)(43 49)(44 48)(45 47)(51 56)(52 60)(53 59)(54 58)(55 57)
G:=sub<Sym(60)| (1,21,11)(2,22,12)(3,23,13)(4,24,14)(5,25,15)(6,26,16)(7,27,17)(8,28,18)(9,29,19)(10,30,20)(31,51,41)(32,52,42)(33,53,43)(34,54,44)(35,55,45)(36,56,46)(37,57,47)(38,58,48)(39,59,49)(40,60,50), (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)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60), (1,31,6,36)(2,35,7,40)(3,34,8,39)(4,33,9,38)(5,32,10,37)(11,41,16,46)(12,45,17,50)(13,44,18,49)(14,43,19,48)(15,42,20,47)(21,51,26,56)(22,55,27,60)(23,54,28,59)(24,53,29,58)(25,52,30,57), (2,5)(3,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(31,36)(32,40)(33,39)(34,38)(35,37)(41,46)(42,50)(43,49)(44,48)(45,47)(51,56)(52,60)(53,59)(54,58)(55,57)>;
G:=Group( (1,21,11)(2,22,12)(3,23,13)(4,24,14)(5,25,15)(6,26,16)(7,27,17)(8,28,18)(9,29,19)(10,30,20)(31,51,41)(32,52,42)(33,53,43)(34,54,44)(35,55,45)(36,56,46)(37,57,47)(38,58,48)(39,59,49)(40,60,50), (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)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60), (1,31,6,36)(2,35,7,40)(3,34,8,39)(4,33,9,38)(5,32,10,37)(11,41,16,46)(12,45,17,50)(13,44,18,49)(14,43,19,48)(15,42,20,47)(21,51,26,56)(22,55,27,60)(23,54,28,59)(24,53,29,58)(25,52,30,57), (2,5)(3,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(31,36)(32,40)(33,39)(34,38)(35,37)(41,46)(42,50)(43,49)(44,48)(45,47)(51,56)(52,60)(53,59)(54,58)(55,57) );
G=PermutationGroup([[(1,21,11),(2,22,12),(3,23,13),(4,24,14),(5,25,15),(6,26,16),(7,27,17),(8,28,18),(9,29,19),(10,30,20),(31,51,41),(32,52,42),(33,53,43),(34,54,44),(35,55,45),(36,56,46),(37,57,47),(38,58,48),(39,59,49),(40,60,50)], [(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),(46,47,48,49,50),(51,52,53,54,55),(56,57,58,59,60)], [(1,31,6,36),(2,35,7,40),(3,34,8,39),(4,33,9,38),(5,32,10,37),(11,41,16,46),(12,45,17,50),(13,44,18,49),(14,43,19,48),(15,42,20,47),(21,51,26,56),(22,55,27,60),(23,54,28,59),(24,53,29,58),(25,52,30,57)], [(2,5),(3,4),(7,10),(8,9),(12,15),(13,14),(17,20),(18,19),(22,25),(23,24),(27,30),(28,29),(31,36),(32,40),(33,39),(34,38),(35,37),(41,46),(42,50),(43,49),(44,48),(45,47),(51,56),(52,60),(53,59),(54,58),(55,57)]])
C3×C5⋊D4 is a maximal subgroup of
Dic5.D6 C30.C23 D10⋊D6 C3×D4×D5 SL2(𝔽3).11D10
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 10A | ··· | 10F | 12A | 12B | 15A | 15B | 15C | 15D | 30A | ··· | 30L |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | 2 | 10 | 1 | 1 | 10 | 2 | 2 | 1 | 1 | 2 | 2 | 10 | 10 | 2 | ··· | 2 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | D5 | D10 | C3×D4 | C3×D5 | C5⋊D4 | C6×D5 | C3×C5⋊D4 |
| kernel | C3×C5⋊D4 | C3×Dic5 | C6×D5 | C2×C30 | C5⋊D4 | Dic5 | D10 | C2×C10 | C15 | C2×C6 | C6 | C5 | C22 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
Matrix representation of C3×C5⋊D4 ►in GL2(𝔽31) generated by
| 25 | 0 |
| 0 | 25 |
| 26 | 12 |
| 29 | 17 |
| 10 | 8 |
| 30 | 21 |
| 1 | 20 |
| 0 | 30 |
G:=sub<GL(2,GF(31))| [25,0,0,25],[26,29,12,17],[10,30,8,21],[1,0,20,30] >;
C3×C5⋊D4 in GAP, Magma, Sage, TeX
C_3\times C_5\rtimes D_4
% in TeX
G:=Group("C3xC5:D4"); // GroupNames label
G:=SmallGroup(120,20);
// by ID
G=gap.SmallGroup(120,20);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-5,141,2404]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^5=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
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