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G = D15:D5order 300 = 22·3·52

The semidirect product of D15 and D5 acting via D5/C5=C2

metabelian, supersoluble, monomial, A-group

Aliases: D15:D5, C52:4D6, C15:2D10, C3:2D52, C5:D5:2S3, C5:3(S3xD5), (C5xD15):3C2, (C5xC15):4C22, (C3xC5:D5):2C2, SmallGroup(300,40)

Series: Derived Chief Lower central Upper central

C1C5xC15 — D15:D5
C1C5C52C5xC15C5xD15 — D15:D5
C5xC15 — D15:D5
C1

Generators and relations for D15:D5
 G = < a,b,c,d | a15=b2=c5=d2=1, bab=a-1, ac=ca, dad=a4, bc=cb, dbd=a3b, dcd=c-1 >

Subgroups: 368 in 48 conjugacy classes, 15 normal (7 characteristic)
Quotients: C1, C2, C22, S3, D5, D6, D10, S3xD5, D52, D15:D5
15C2
15C2
25C2
2C5
2C5
75C22
5S3
5S3
25C6
3D5
3D5
5D5
5D5
10D5
10D5
15C10
15C10
2C15
2C15
25D6
15D10
15D10
5C5xS3
5C3xD5
5C3xD5
5C5xS3
10C3xD5
10C3xD5
3C5xD5
3C5xD5
5S3xD5
5S3xD5
3D52

Character table of D15:D5

 class 12A2B2C35A5B5C5D5E5F5G5H610A10B10C10D15A15B15C15D15E15F15G15H15I15J15K15L
 size 11515252222244445030303030444444444444
ρ1111111111111111111111111111111    trivial
ρ211-1-1111111111-11-11-1111111111111    linear of order 2
ρ31-11-1111111111-1-11-11111111111111    linear of order 2
ρ41-1-111111111111-1-1-1-1111111111111    linear of order 2
ρ52002-122222222-10000-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ6200-2-12222222210000-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from D6
ρ720-20222-1+5/2-1-5/2-1-5/2-1+5/2-1+5/2-1-5/2001-5/201+5/2-1+5/2-1-5/2-1-5/2-1+5/2-1+5/2-1-5/22-1+5/2-1-5/2-1-5/2-1+5/22    orthogonal lifted from D10
ρ82-2002-1-5/2-1+5/222-1-5/2-1-5/2-1+5/2-1+5/201-5/201+5/20-1+5/2-1+5/2-1+5/2-1+5/222-1-5/2-1-5/2-1-5/2-1-5/2-1-5/2-1+5/2    orthogonal lifted from D10
ρ92-2002-1+5/2-1-5/222-1+5/2-1+5/2-1-5/2-1-5/201+5/201-5/20-1-5/2-1-5/2-1-5/2-1-5/222-1+5/2-1+5/2-1+5/2-1+5/2-1+5/2-1-5/2    orthogonal lifted from D10
ρ1022002-1+5/2-1-5/222-1+5/2-1+5/2-1-5/2-1-5/20-1-5/20-1+5/20-1-5/2-1-5/2-1-5/2-1-5/222-1+5/2-1+5/2-1+5/2-1+5/2-1+5/2-1-5/2    orthogonal lifted from D5
ρ112020222-1+5/2-1-5/2-1-5/2-1+5/2-1+5/2-1-5/200-1+5/20-1-5/2-1+5/2-1-5/2-1-5/2-1+5/2-1+5/2-1-5/22-1+5/2-1-5/2-1-5/2-1+5/22    orthogonal lifted from D5
ρ1220-20222-1-5/2-1+5/2-1+5/2-1-5/2-1-5/2-1+5/2001+5/201-5/2-1-5/2-1+5/2-1+5/2-1-5/2-1-5/2-1+5/22-1-5/2-1+5/2-1+5/2-1-5/22    orthogonal lifted from D10
ρ132020222-1-5/2-1+5/2-1+5/2-1-5/2-1-5/2-1+5/200-1-5/20-1+5/2-1-5/2-1+5/2-1+5/2-1-5/2-1-5/2-1+5/22-1-5/2-1+5/2-1+5/2-1-5/22    orthogonal lifted from D5
ρ1422002-1-5/2-1+5/222-1-5/2-1-5/2-1+5/2-1+5/20-1+5/20-1-5/20-1+5/2-1+5/2-1+5/2-1+5/222-1-5/2-1-5/2-1-5/2-1-5/2-1-5/2-1+5/2    orthogonal lifted from D5
ρ154000-2-1+5-1-544-1+5-1+5-1-5-1-5000001+5/21+5/21+5/21+5/2-2-21-5/21-5/21-5/21-5/21-5/21+5/2    orthogonal lifted from S3xD5
ρ1640004-1+5-1-5-1+5-1-5-13-5/2-13+5/200000-13+5/23+5/2-1-1+5-1-5-1+53-5/2-1-13-5/2-1-5    orthogonal lifted from D52
ρ174000-244-1+5-1-5-1-5-1+5-1+5-1-5000001-5/21+5/21+5/21-5/21-5/21+5/2-21-5/21+5/21+5/21-5/2-2    orthogonal lifted from S3xD5
ρ1840004-1+5-1-5-1-5-1+53-5/2-13+5/2-1000003+5/2-1-13+5/2-1-5-1+5-1+5-13-5/23-5/2-1-1-5    orthogonal lifted from D52
ρ194000-244-1-5-1+5-1+5-1-5-1-5-1+5000001+5/21-5/21-5/21+5/21+5/21-5/2-21+5/21-5/21-5/21+5/2-2    orthogonal lifted from S3xD5
ρ2040004-1-5-1+5-1+5-1-53+5/2-13-5/2-1000003-5/2-1-13-5/2-1+5-1-5-1-5-13+5/23+5/2-1-1+5    orthogonal lifted from D52
ρ2140004-1-5-1+5-1-5-1+5-13+5/2-13-5/200000-13-5/23-5/2-1-1-5-1+5-1-53+5/2-1-13+5/2-1+5    orthogonal lifted from D52
ρ224000-2-1-5-1+544-1-5-1-5-1+5-1+5000001-5/21-5/21-5/21-5/2-2-21+5/21+5/21+5/21+5/21+5/21-5/2    orthogonal lifted from S3xD5
ρ234000-2-1+5-1-5-1-5-1+53-5/2-13+5/2-100000ζ3ζ543ζ5-2ζ3-21+-15/21--15/2ζ32ζ5432ζ5-2ζ32-21+5/21-5/21-5/21--15/2ζ32ζ5332ζ52-2ζ32-2ζ3ζ533ζ52-2ζ3-21+-15/21+5/2    complex faithful
ρ244000-2-1+5-1-5-1+5-1-5-13-5/2-13+5/2000001+-15/2ζ32ζ5432ζ5-2ζ32-2ζ3ζ543ζ5-2ζ3-21--15/21-5/21+5/21-5/2ζ32ζ5332ζ52-2ζ32-21+-15/21--15/2ζ3ζ533ζ52-2ζ3-21+5/2    complex faithful
ρ254000-2-1-5-1+5-1-5-1+5-13+5/2-13-5/2000001--15/2ζ32ζ5332ζ52-2ζ32-2ζ3ζ533ζ52-2ζ3-21+-15/21+5/21-5/21+5/2ζ32ζ5432ζ5-2ζ32-21--15/21+-15/2ζ3ζ543ζ5-2ζ3-21-5/2    complex faithful
ρ264000-2-1-5-1+5-1+5-1-53+5/2-13-5/2-100000ζ32ζ5332ζ52-2ζ32-21+-15/21--15/2ζ3ζ533ζ52-2ζ3-21-5/21+5/21+5/21--15/2ζ3ζ543ζ5-2ζ3-2ζ32ζ5432ζ5-2ζ32-21+-15/21-5/2    complex faithful
ρ274000-2-1-5-1+5-1-5-1+5-13+5/2-13-5/2000001+-15/2ζ3ζ533ζ52-2ζ3-2ζ32ζ5332ζ52-2ζ32-21--15/21+5/21-5/21+5/2ζ3ζ543ζ5-2ζ3-21+-15/21--15/2ζ32ζ5432ζ5-2ζ32-21-5/2    complex faithful
ρ284000-2-1-5-1+5-1+5-1-53+5/2-13-5/2-100000ζ3ζ533ζ52-2ζ3-21--15/21+-15/2ζ32ζ5332ζ52-2ζ32-21-5/21+5/21+5/21+-15/2ζ32ζ5432ζ5-2ζ32-2ζ3ζ543ζ5-2ζ3-21--15/21-5/2    complex faithful
ρ294000-2-1+5-1-5-1+5-1-5-13-5/2-13+5/2000001--15/2ζ3ζ543ζ5-2ζ3-2ζ32ζ5432ζ5-2ζ32-21+-15/21-5/21+5/21-5/2ζ3ζ533ζ52-2ζ3-21--15/21+-15/2ζ32ζ5332ζ52-2ζ32-21+5/2    complex faithful
ρ304000-2-1+5-1-5-1-5-1+53-5/2-13+5/2-100000ζ32ζ5432ζ5-2ζ32-21--15/21+-15/2ζ3ζ543ζ5-2ζ3-21+5/21-5/21-5/21+-15/2ζ3ζ533ζ52-2ζ3-2ζ32ζ5332ζ52-2ζ32-21--15/21+5/2    complex faithful

Permutation representations of D15:D5
On 30 points - transitive group 30T79
Generators in S30
(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 25)(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 30)(12 29)(13 28)(14 27)(15 26)
(1 7 13 4 10)(2 8 14 5 11)(3 9 15 6 12)(16 25 19 28 22)(17 26 20 29 23)(18 27 21 30 24)
(1 10)(2 14)(4 7)(5 11)(6 15)(9 12)(16 22)(17 26)(18 30)(20 23)(21 27)(25 28)

G:=sub<Sym(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), (1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,30)(12,29)(13,28)(14,27)(15,26), (1,7,13,4,10)(2,8,14,5,11)(3,9,15,6,12)(16,25,19,28,22)(17,26,20,29,23)(18,27,21,30,24), (1,10)(2,14)(4,7)(5,11)(6,15)(9,12)(16,22)(17,26)(18,30)(20,23)(21,27)(25,28)>;

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), (1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,30)(12,29)(13,28)(14,27)(15,26), (1,7,13,4,10)(2,8,14,5,11)(3,9,15,6,12)(16,25,19,28,22)(17,26,20,29,23)(18,27,21,30,24), (1,10)(2,14)(4,7)(5,11)(6,15)(9,12)(16,22)(17,26)(18,30)(20,23)(21,27)(25,28) );

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)], [(1,25),(2,24),(3,23),(4,22),(5,21),(6,20),(7,19),(8,18),(9,17),(10,16),(11,30),(12,29),(13,28),(14,27),(15,26)], [(1,7,13,4,10),(2,8,14,5,11),(3,9,15,6,12),(16,25,19,28,22),(17,26,20,29,23),(18,27,21,30,24)], [(1,10),(2,14),(4,7),(5,11),(6,15),(9,12),(16,22),(17,26),(18,30),(20,23),(21,27),(25,28)]])

G:=TransitiveGroup(30,79);

Matrix representation of D15:D5 in GL6(F31)

3010000
3000000
00301300
00181300
000010
000001
,
1740000
21140000
00131800
0011800
0000300
0000030
,
100000
010000
001000
000100
0000030
0000112
,
100000
010000
000100
001000
0000112
0000030

G:=sub<GL(6,GF(31))| [30,30,0,0,0,0,1,0,0,0,0,0,0,0,30,18,0,0,0,0,13,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[17,21,0,0,0,0,4,14,0,0,0,0,0,0,13,1,0,0,0,0,18,18,0,0,0,0,0,0,30,0,0,0,0,0,0,30],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,30,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,12,30] >;

D15:D5 in GAP, Magma, Sage, TeX

D_{15}\rtimes D_5
% in TeX

G:=Group("D15:D5");
// GroupNames label

G:=SmallGroup(300,40);
// by ID

G=gap.SmallGroup(300,40);
# by ID

G:=PCGroup([5,-2,-2,-3,-5,-5,122,963,488,3009]);
// Polycyclic

G:=Group<a,b,c,d|a^15=b^2=c^5=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^4,b*c=c*b,d*b*d=a^3*b,d*c*d=c^-1>;
// generators/relations

Export

Subgroup lattice of D15:D5 in TeX
Character table of D15:D5 in TeX

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