non-abelian, soluble, monomial
Aliases: A4⋊D15, C15⋊1S4, C3⋊(C5⋊S4), C5⋊(C3⋊S4), (C5×A4)⋊1S3, (C2×C30)⋊2S3, (C3×A4)⋊2D5, (C2×C6)⋊2D15, (A4×C15)⋊2C2, C22⋊(C3⋊D15), (C2×C10)⋊2(C3⋊S3), SmallGroup(360,141)
Series: Derived ►Chief ►Lower central ►Upper central
| A4×C15 — A4⋊D15 |
Generators and relations for A4⋊D15
G = < a,b,c,d,e | a2=b2=c3=d15=e2=1, cac-1=ebe=ab=ba, ad=da, ae=ea, cbc-1=a, bd=db, cd=dc, ece=c-1, ede=d-1 >
Subgroups: 720 in 60 conjugacy classes, 17 normal (11 characteristic)
C1, C2, C3, C3, C4, C22, C22, C5, S3, C6, D4, C32, D5, C10, Dic3, A4, D6, C2×C6, C15, C15, C3⋊S3, Dic5, D10, C2×C10, C3⋊D4, S4, D15, C30, C3×A4, C5⋊D4, C3×C15, Dic15, C5×A4, D30, C2×C30, C3⋊S4, C3⋊D15, C15⋊7D4, C5⋊S4, A4×C15, A4⋊D15
Quotients: C1, C2, S3, D5, C3⋊S3, S4, D15, C3⋊S4, C3⋊D15, C5⋊S4, A4⋊D15
►On 60 points
Generators in S60
(1 27)(2 28)(3 29)(4 30)(5 16)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(12 23)(13 24)(14 25)(15 26)(31 57)(32 58)(33 59)(34 60)(35 46)(36 47)(37 48)(38 49)(39 50)(40 51)(41 52)(42 53)(43 54)(44 55)(45 56)
(1 44)(2 45)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 59)(17 60)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)
(16 33 59)(17 34 60)(18 35 46)(19 36 47)(20 37 48)(21 38 49)(22 39 50)(23 40 51)(24 41 52)(25 42 53)(26 43 54)(27 44 55)(28 45 56)(29 31 57)(30 32 58)
(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 22)(17 21)(18 20)(23 30)(24 29)(25 28)(26 27)(31 52)(32 51)(33 50)(34 49)(35 48)(36 47)(37 46)(38 60)(39 59)(40 58)(41 57)(42 56)(43 55)(44 54)(45 53)
G:=sub<Sym(60)| (1,27)(2,28)(3,29)(4,30)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(12,23)(13,24)(14,25)(15,26)(31,57)(32,58)(33,59)(34,60)(35,46)(36,47)(37,48)(38,49)(39,50)(40,51)(41,52)(42,53)(43,54)(44,55)(45,56), (1,44)(2,45)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,59)(17,60)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58), (16,33,59)(17,34,60)(18,35,46)(19,36,47)(20,37,48)(21,38,49)(22,39,50)(23,40,51)(24,41,52)(25,42,53)(26,43,54)(27,44,55)(28,45,56)(29,31,57)(30,32,58), (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)(16,22)(17,21)(18,20)(23,30)(24,29)(25,28)(26,27)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,60)(39,59)(40,58)(41,57)(42,56)(43,55)(44,54)(45,53)>;
G:=Group( (1,27)(2,28)(3,29)(4,30)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(12,23)(13,24)(14,25)(15,26)(31,57)(32,58)(33,59)(34,60)(35,46)(36,47)(37,48)(38,49)(39,50)(40,51)(41,52)(42,53)(43,54)(44,55)(45,56), (1,44)(2,45)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,59)(17,60)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58), (16,33,59)(17,34,60)(18,35,46)(19,36,47)(20,37,48)(21,38,49)(22,39,50)(23,40,51)(24,41,52)(25,42,53)(26,43,54)(27,44,55)(28,45,56)(29,31,57)(30,32,58), (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)(16,22)(17,21)(18,20)(23,30)(24,29)(25,28)(26,27)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,60)(39,59)(40,58)(41,57)(42,56)(43,55)(44,54)(45,53) );
G=PermutationGroup([[(1,27),(2,28),(3,29),(4,30),(5,16),(6,17),(7,18),(8,19),(9,20),(10,21),(11,22),(12,23),(13,24),(14,25),(15,26),(31,57),(32,58),(33,59),(34,60),(35,46),(36,47),(37,48),(38,49),(39,50),(40,51),(41,52),(42,53),(43,54),(44,55),(45,56)], [(1,44),(2,45),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(15,43),(16,59),(17,60),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(29,57),(30,58)], [(16,33,59),(17,34,60),(18,35,46),(19,36,47),(20,37,48),(21,38,49),(22,39,50),(23,40,51),(24,41,52),(25,42,53),(26,43,54),(27,44,55),(28,45,56),(29,31,57),(30,32,58)], [(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),(16,22),(17,21),(18,20),(23,30),(24,29),(25,28),(26,27),(31,52),(32,51),(33,50),(34,49),(35,48),(36,47),(37,46),(38,60),(39,59),(40,58),(41,57),(42,56),(43,55),(44,54),(45,53)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 4 | 5A | 5B | 6 | 10A | 10B | 15A | 15B | 15C | 15D | 15E | ··· | 15P | 30A | 30B | 30C | 30D |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 5 | 5 | 6 | 10 | 10 | 15 | 15 | 15 | 15 | 15 | ··· | 15 | 30 | 30 | 30 | 30 |
| size | 1 | 3 | 90 | 2 | 8 | 8 | 8 | 90 | 2 | 2 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 8 | ··· | 8 | 6 | 6 | 6 | 6 |
33 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 6 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | S3 | S3 | D5 | D15 | D15 | S4 | C3⋊S4 | C5⋊S4 | A4⋊D15 |
| kernel | A4⋊D15 | A4×C15 | C5×A4 | C2×C30 | C3×A4 | A4 | C2×C6 | C15 | C5 | C3 | C1 |
| # reps | 1 | 1 | 3 | 1 | 2 | 12 | 4 | 2 | 1 | 2 | 4 |
Matrix representation of A4⋊D15 ►in GL5(𝔽61)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 60 |
| 0 | 0 | 60 | 0 | 1 |
| 0 | 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 | 60 |
| 0 | 0 | 60 | 60 | 0 |
| 0 | 60 | 0 | 0 | 0 |
| 1 | 60 | 0 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 | 1 |
| 30 | 12 | 0 | 0 | 0 |
| 49 | 42 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 34 | 51 | 0 | 0 | 0 |
| 24 | 27 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 |
G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,60,0,0,0,0,60,1,60],[1,0,0,0,0,0,1,0,0,0,0,0,60,1,60,0,0,0,0,60,0,0,0,60,0],[0,1,0,0,0,60,60,0,0,0,0,0,60,1,60,0,0,60,0,0,0,0,0,0,1],[30,49,0,0,0,12,42,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[34,24,0,0,0,51,27,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,60] >;
A4⋊D15 in GAP, Magma, Sage, TeX
A_4\rtimes D_{15} % in TeX
G:=Group("A4:D15"); // GroupNames label
G:=SmallGroup(360,141);
// by ID
G=gap.SmallGroup(360,141);
# by ID
G:=PCGroup([6,-2,-3,-3,-5,-2,2,49,218,1731,5404,4060,3245,1631]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^15=e^2=1,c*a*c^-1=e*b*e=a*b=b*a,a*d=d*a,a*e=e*a,c*b*c^-1=a,b*d=d*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations