non-abelian, soluble, monomial
Aliases: C24⋊D15, C24⋊C5⋊S3, C3⋊(C24⋊D5), (C23×C6)⋊2D5, (C3×C24⋊C5)⋊1C2, SmallGroup(480,1195)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C24⋊C5 — C24⋊D15 |
Generators and relations for C24⋊D15
G = < a,b,c,d,e,f | a2=b2=c2=d2=e15=f2=1, eae-1=ab=ba, faf=ac=ca, ad=da, ebe-1=fbf=bc=cb, bd=db, ece-1=cd=dc, cf=fc, ede-1=a, fdf=abcd, fef=e-1 >
Subgroups: 1016 in 86 conjugacy classes, 7 normal (all characteristic)
C1, C2, C3, C4, C22, C5, S3, C6, C2×C4, D4, C23, D5, Dic3, D6, C2×C6, C15, C22⋊C4, C2×D4, C24, C2×Dic3, C3⋊D4, C22×S3, C22×C6, D15, C22≀C2, C6.D4, C2×C3⋊D4, C23×C6, C24⋊C5, C24⋊4S3, C24⋊D5, C3×C24⋊C5, C24⋊D15
Quotients: C1, C2, S3, D5, D15, C24⋊D5, C24⋊D15
Character table of C24⋊D15
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 15A | 15B | 15C | 15D | |
| size | 1 | 5 | 5 | 5 | 60 | 2 | 60 | 60 | 60 | 32 | 32 | 10 | 10 | 10 | 32 | 32 | 32 | 32 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 2 | 2 | 2 | 2 | 0 | -1 | 0 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ4 | 2 | 2 | 2 | 2 | 0 | 2 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 2 | 2 | 2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| ρ5 | 2 | 2 | 2 | 2 | 0 | 2 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 2 | 2 | 2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | orthogonal lifted from D5 |
| ρ6 | 2 | 2 | 2 | 2 | 0 | -1 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1 | -1 | -1 | -ζ32ζ54+ζ32ζ5-ζ54 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ3ζ53+ζ3ζ52-ζ53 | -ζ3ζ54+ζ3ζ5-ζ54 | orthogonal lifted from D15 |
| ρ7 | 2 | 2 | 2 | 2 | 0 | -1 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1 | -1 | -1 | -ζ3ζ53+ζ3ζ52-ζ53 | -ζ32ζ54+ζ32ζ5-ζ54 | -ζ3ζ54+ζ3ζ5-ζ54 | ζ3ζ53-ζ3ζ52-ζ52 | orthogonal lifted from D15 |
| ρ8 | 2 | 2 | 2 | 2 | 0 | -1 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1 | -1 | -1 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ3ζ54+ζ3ζ5-ζ54 | -ζ32ζ54+ζ32ζ5-ζ54 | -ζ3ζ53+ζ3ζ52-ζ53 | orthogonal lifted from D15 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | -1 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1 | -1 | -1 | -ζ3ζ54+ζ3ζ5-ζ54 | -ζ3ζ53+ζ3ζ52-ζ53 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ32ζ54+ζ32ζ5-ζ54 | orthogonal lifted from D15 |
| ρ10 | 5 | 1 | -3 | 1 | -1 | 5 | -1 | 1 | 1 | 0 | 0 | 1 | 1 | -3 | 0 | 0 | 0 | 0 | orthogonal lifted from C24⋊D5 |
| ρ11 | 5 | 1 | -3 | 1 | 1 | 5 | 1 | -1 | -1 | 0 | 0 | 1 | 1 | -3 | 0 | 0 | 0 | 0 | orthogonal lifted from C24⋊D5 |
| ρ12 | 5 | 1 | 1 | -3 | 1 | 5 | -1 | -1 | 1 | 0 | 0 | 1 | -3 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from C24⋊D5 |
| ρ13 | 5 | 1 | 1 | -3 | -1 | 5 | 1 | 1 | -1 | 0 | 0 | 1 | -3 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from C24⋊D5 |
| ρ14 | 5 | -3 | 1 | 1 | 1 | 5 | -1 | 1 | -1 | 0 | 0 | -3 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from C24⋊D5 |
| ρ15 | 5 | -3 | 1 | 1 | -1 | 5 | 1 | -1 | 1 | 0 | 0 | -3 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from C24⋊D5 |
| ρ16 | 10 | 2 | 2 | -6 | 0 | -5 | 0 | 0 | 0 | 0 | 0 | -1 | 3 | -1 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ17 | 10 | -6 | 2 | 2 | 0 | -5 | 0 | 0 | 0 | 0 | 0 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ18 | 10 | 2 | -6 | 2 | 0 | -5 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 3 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 30 points - transitive group 30T106
Generators in S30
(3 19)(5 21)(8 24)(10 26)(13 29)(15 16)
(2 18)(3 19)(4 20)(5 21)(7 23)(8 24)(9 25)(10 26)(12 28)(13 29)(14 30)(15 16)
(1 17)(5 21)(6 22)(10 26)(11 27)(15 16)
(1 17)(4 20)(6 22)(9 25)(11 27)(14 30)
(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)
(1 16)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 24)(9 23)(10 22)(11 21)(12 20)(13 19)(14 18)(15 17)
G:=sub<Sym(30)| (3,19)(5,21)(8,24)(10,26)(13,29)(15,16), (2,18)(3,19)(4,20)(5,21)(7,23)(8,24)(9,25)(10,26)(12,28)(13,29)(14,30)(15,16), (1,17)(5,21)(6,22)(10,26)(11,27)(15,16), (1,17)(4,20)(6,22)(9,25)(11,27)(14,30), (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), (1,16)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)>;
G:=Group( (3,19)(5,21)(8,24)(10,26)(13,29)(15,16), (2,18)(3,19)(4,20)(5,21)(7,23)(8,24)(9,25)(10,26)(12,28)(13,29)(14,30)(15,16), (1,17)(5,21)(6,22)(10,26)(11,27)(15,16), (1,17)(4,20)(6,22)(9,25)(11,27)(14,30), (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), (1,16)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17) );
G=PermutationGroup([[(3,19),(5,21),(8,24),(10,26),(13,29),(15,16)], [(2,18),(3,19),(4,20),(5,21),(7,23),(8,24),(9,25),(10,26),(12,28),(13,29),(14,30),(15,16)], [(1,17),(5,21),(6,22),(10,26),(11,27),(15,16)], [(1,17),(4,20),(6,22),(9,25),(11,27),(14,30)], [(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)], [(1,16),(2,30),(3,29),(4,28),(5,27),(6,26),(7,25),(8,24),(9,23),(10,22),(11,21),(12,20),(13,19),(14,18),(15,17)]])
G:=TransitiveGroup(30,106);
►On 30 points - transitive group 30T119
Generators in S30
(3 19)(5 21)(8 24)(10 26)(13 29)(15 16)
(2 18)(3 19)(4 20)(5 21)(7 23)(8 24)(9 25)(10 26)(12 28)(13 29)(14 30)(15 16)
(1 17)(5 21)(6 22)(10 26)(11 27)(15 16)
(1 17)(4 20)(6 22)(9 25)(11 27)(14 30)
(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)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 17)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)
G:=sub<Sym(30)| (3,19)(5,21)(8,24)(10,26)(13,29)(15,16), (2,18)(3,19)(4,20)(5,21)(7,23)(8,24)(9,25)(10,26)(12,28)(13,29)(14,30)(15,16), (1,17)(5,21)(6,22)(10,26)(11,27)(15,16), (1,17)(4,20)(6,22)(9,25)(11,27)(14,30), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,17)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)>;
G:=Group( (3,19)(5,21)(8,24)(10,26)(13,29)(15,16), (2,18)(3,19)(4,20)(5,21)(7,23)(8,24)(9,25)(10,26)(12,28)(13,29)(14,30)(15,16), (1,17)(5,21)(6,22)(10,26)(11,27)(15,16), (1,17)(4,20)(6,22)(9,25)(11,27)(14,30), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,17)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25) );
G=PermutationGroup([[(3,19),(5,21),(8,24),(10,26),(13,29),(15,16)], [(2,18),(3,19),(4,20),(5,21),(7,23),(8,24),(9,25),(10,26),(12,28),(13,29),(14,30),(15,16)], [(1,17),(5,21),(6,22),(10,26),(11,27),(15,16)], [(1,17),(4,20),(6,22),(9,25),(11,27),(14,30)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,17),(18,30),(19,29),(20,28),(21,27),(22,26),(23,25)]])
G:=TransitiveGroup(30,119);
Matrix representation of C24⋊D15 ►in GL7(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 60 |
| 23 | 14 | 0 | 0 | 0 | 0 | 0 |
| 39 | 45 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 60 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(7,GF(61))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60],[23,39,0,0,0,0,0,14,45,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0],[60,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0] >;
C24⋊D15 in GAP, Magma, Sage, TeX
C_2^4\rtimes D_{15} % in TeX
G:=Group("C2^4:D15"); // GroupNames label
G:=SmallGroup(480,1195);
// by ID
G=gap.SmallGroup(480,1195);
# by ID
G:=PCGroup([7,-2,-3,-5,-2,2,2,2,57,506,2523,717,1768,13865,2749,7356,755]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^15=f^2=1,e*a*e^-1=a*b=b*a,f*a*f=a*c=c*a,a*d=d*a,e*b*e^-1=f*b*f=b*c=c*b,b*d=d*b,e*c*e^-1=c*d=d*c,c*f=f*c,e*d*e^-1=a,f*d*f=a*b*c*d,f*e*f=e^-1>;
// generators/relations
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