direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4×D15, C4⋊1D30, C20⋊3D6, D60⋊3C2, C12⋊3D10, C60⋊1C22, C22⋊2D30, D30⋊6C22, C30.32C23, Dic15⋊3C22, C5⋊4(S3×D4), C3⋊4(D4×D5), (C5×D4)⋊2S3, (C3×D4)⋊2D5, (C2×C10)⋊6D6, (C2×C6)⋊3D10, C15⋊13(C2×D4), (D4×C15)⋊2C2, (C4×D15)⋊1C2, C15⋊7D4⋊1C2, (C2×C30)⋊1C22, (C22×D15)⋊2C2, C6.32(C22×D5), C2.6(C22×D15), C10.32(C22×S3), SmallGroup(240,179)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×D15
G = < a,b,c,d | a4=b2=c15=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 640 in 108 conjugacy classes, 37 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, D4, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C2×D4, Dic5, C20, D10, C2×C10, C4×S3, D12, C3⋊D4, C3×D4, C22×S3, D15, D15, C30, C30, C4×D5, D20, C5⋊D4, C5×D4, C22×D5, S3×D4, Dic15, C60, D30, D30, D30, C2×C30, D4×D5, C4×D15, D60, C15⋊7D4, D4×C15, C22×D15, D4×D15
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, D15, C22×D5, S3×D4, D30, D4×D5, C22×D15, D4×D15
►On 60 points
(1 58 16 43)(2 59 17 44)(3 60 18 45)(4 46 19 31)(5 47 20 32)(6 48 21 33)(7 49 22 34)(8 50 23 35)(9 51 24 36)(10 52 25 37)(11 53 26 38)(12 54 27 39)(13 55 28 40)(14 56 29 41)(15 57 30 42)
(1 43)(2 44)(3 45)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 58)(17 59)(18 60)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 30)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)(31 39)(32 38)(33 37)(34 36)(40 45)(41 44)(42 43)(46 54)(47 53)(48 52)(49 51)(55 60)(56 59)(57 58)
G:=sub<Sym(60)| (1,58,16,43)(2,59,17,44)(3,60,18,45)(4,46,19,31)(5,47,20,32)(6,48,21,33)(7,49,22,34)(8,50,23,35)(9,51,24,36)(10,52,25,37)(11,53,26,38)(12,54,27,39)(13,55,28,40)(14,56,29,41)(15,57,30,42), (1,43)(2,44)(3,45)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,58)(17,59)(18,60)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,30)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)(31,39)(32,38)(33,37)(34,36)(40,45)(41,44)(42,43)(46,54)(47,53)(48,52)(49,51)(55,60)(56,59)(57,58)>;
G:=Group( (1,58,16,43)(2,59,17,44)(3,60,18,45)(4,46,19,31)(5,47,20,32)(6,48,21,33)(7,49,22,34)(8,50,23,35)(9,51,24,36)(10,52,25,37)(11,53,26,38)(12,54,27,39)(13,55,28,40)(14,56,29,41)(15,57,30,42), (1,43)(2,44)(3,45)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,58)(17,59)(18,60)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,30)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)(31,39)(32,38)(33,37)(34,36)(40,45)(41,44)(42,43)(46,54)(47,53)(48,52)(49,51)(55,60)(56,59)(57,58) );
G=PermutationGroup([[(1,58,16,43),(2,59,17,44),(3,60,18,45),(4,46,19,31),(5,47,20,32),(6,48,21,33),(7,49,22,34),(8,50,23,35),(9,51,24,36),(10,52,25,37),(11,53,26,38),(12,54,27,39),(13,55,28,40),(14,56,29,41),(15,57,30,42)], [(1,43),(2,44),(3,45),(4,31),(5,32),(6,33),(7,34),(8,35),(9,36),(10,37),(11,38),(12,39),(13,40),(14,41),(15,42),(16,58),(17,59),(18,60),(19,46),(20,47),(21,48),(22,49),(23,50),(24,51),(25,52),(26,53),(27,54),(28,55),(29,56),(30,57)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,30),(17,29),(18,28),(19,27),(20,26),(21,25),(22,24),(31,39),(32,38),(33,37),(34,36),(40,45),(41,44),(42,43),(46,54),(47,53),(48,52),(49,51),(55,60),(56,59),(57,58)]])
D4×D15 is a maximal subgroup of
D15⋊D8 Dic10⋊D6 Dic6⋊D10 D12⋊5D10 D8⋊D15 Q8⋊3D30 S3×D4×D5 D30.C23 D20⋊14D6 D12⋊14D10 D4⋊6D30 D4⋊8D30
D4×D15 is a maximal quotient of
C22⋊2Dic30 Dic15⋊19D4 D30⋊16D4 D30.28D4 D30⋊9D4 C23.11D30 C4⋊Dic30 D60⋊11C4 D30.29D4 C4⋊D60 D30⋊5Q8 D8⋊D15 D8⋊3D15 Q8⋊3D30 SD16⋊D15 D4.5D30 Q16⋊D15 D120⋊8C2 D30⋊17D4 C60⋊2D4 Dic15⋊12D4 C60⋊3D4
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 5A | 5B | 6A | 6B | 6C | 10A | 10B | 10C | 10D | 10E | 10F | 12 | 15A | 15B | 15C | 15D | 20A | 20B | 30A | 30B | 30C | 30D | 30E | ··· | 30L | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 2 | 2 | 15 | 15 | 30 | 30 | 2 | 2 | 30 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D6 | D10 | D10 | D15 | D30 | D30 | S3×D4 | D4×D5 | D4×D15 |
| kernel | D4×D15 | C4×D15 | D60 | C15⋊7D4 | D4×C15 | C22×D15 | C5×D4 | D15 | C3×D4 | C20 | C2×C10 | C12 | C2×C6 | D4 | C4 | C22 | C5 | C3 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 1 | 2 | 4 |
Matrix representation of D4×D15 ►in GL4(𝔽61) generated by
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 31 | 14 | 0 | 0 |
| 47 | 37 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 28 | 25 | 0 | 0 |
| 37 | 33 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,0,60,0,0,1,0],[60,0,0,0,0,60,0,0,0,0,0,1,0,0,1,0],[31,47,0,0,14,37,0,0,0,0,1,0,0,0,0,1],[28,37,0,0,25,33,0,0,0,0,1,0,0,0,0,1] >;
D4×D15 in GAP, Magma, Sage, TeX
D_4\times D_{15} % in TeX
G:=Group("D4xD15"); // GroupNames label
G:=SmallGroup(240,179);
// by ID
G=gap.SmallGroup(240,179);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,116,964,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^15=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations