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G = C24⋊D15  order 480 = 25·3·5

The semidirect product of C24 and D15 acting via D15/C3=D5

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

Derived series
C1 — C24 — C3×C24⋊C5 — C24⋊D15
Chief series
C1 — C24 — C24⋊C5 — C3×C24⋊C5 — C24⋊D15

Subgroups: 1016 in 86 conjugacy classes, 7 normal (all characteristic)
C1, C2 [×4], C3, C4 [×3], C22 [×10], C5, S3, C6 [×3], C2×C4 [×3], D4 [×6], C23 [×4], D5, Dic3 [×3], D6 [×3], C2×C6 [×7], C15, C22⋊C4 [×3], C2×D4 [×3], C24, C2×Dic3 [×3], C3⋊D4 [×6], C22×S3, C22×C6 [×3], D15, C22≀C2, C6.D4 [×3], C2×C3⋊D4 [×3], C23×C6, C24⋊C5, C24⋊4S3, C24⋊D5, C3×C24⋊C5, C24⋊D15

Quotients:
C1, C2, S3, D5, D15, C24⋊D5, C24⋊D15

Generators and relations
 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 >

Permutation representations
►On 30 points - transitive group 30T106
Generators in S30
(3 30)(5 17)(8 20)(10 22)(13 25)(15 27)

(2 29)(3 30)(4 16)(5 17)(7 19)(8 20)(9 21)(10 22)(12 24)(13 25)(14 26)(15 27)

(1 28)(5 17)(6 18)(10 22)(11 23)(15 27)

(1 28)(4 16)(6 18)(9 21)(11 23)(14 26)

(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 27)(2 26)(3 25)(4 24)(5 23)(6 22)(7 21)(8 20)(9 19)(10 18)(11 17)(12 16)(13 30)(14 29)(15 28)
 
G:=sub<Sym(30)| (3,30)(5,17)(8,20)(10,22)(13,25)(15,27), (2,29)(3,30)(4,16)(5,17)(7,19)(8,20)(9,21)(10,22)(12,24)(13,25)(14,26)(15,27), (1,28)(5,17)(6,18)(10,22)(11,23)(15,27), (1,28)(4,16)(6,18)(9,21)(11,23)(14,26), (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), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(13,30)(14,29)(15,28)>;
 
G:=Group( (3,30)(5,17)(8,20)(10,22)(13,25)(15,27), (2,29)(3,30)(4,16)(5,17)(7,19)(8,20)(9,21)(10,22)(12,24)(13,25)(14,26)(15,27), (1,28)(5,17)(6,18)(10,22)(11,23)(15,27), (1,28)(4,16)(6,18)(9,21)(11,23)(14,26), (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), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(13,30)(14,29)(15,28) );
 
G=PermutationGroup([(3,30),(5,17),(8,20),(10,22),(13,25),(15,27)], [(2,29),(3,30),(4,16),(5,17),(7,19),(8,20),(9,21),(10,22),(12,24),(13,25),(14,26),(15,27)], [(1,28),(5,17),(6,18),(10,22),(11,23),(15,27)], [(1,28),(4,16),(6,18),(9,21),(11,23),(14,26)], [(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)], [(1,27),(2,26),(3,25),(4,24),(5,23),(6,22),(7,21),(8,20),(9,19),(10,18),(11,17),(12,16),(13,30),(14,29),(15,28)])
 
G:=TransitiveGroup(30,106);
 

►On 30 points - transitive group 30T119
Generators in S30
(3 27)(5 29)(8 17)(10 19)(13 22)(15 24)

(2 26)(3 27)(4 28)(5 29)(7 16)(8 17)(9 18)(10 19)(12 21)(13 22)(14 23)(15 24)

(1 25)(5 29)(6 30)(10 19)(11 20)(15 24)

(1 25)(4 28)(6 30)(9 18)(11 20)(14 23)

(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 18)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)
 
G:=sub<Sym(30)| (3,27)(5,29)(8,17)(10,19)(13,22)(15,24), (2,26)(3,27)(4,28)(5,29)(7,16)(8,17)(9,18)(10,19)(12,21)(13,22)(14,23)(15,24), (1,25)(5,29)(6,30)(10,19)(11,20)(15,24), (1,25)(4,28)(6,30)(9,18)(11,20)(14,23), (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)(16,18)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)>;
 
G:=Group( (3,27)(5,29)(8,17)(10,19)(13,22)(15,24), (2,26)(3,27)(4,28)(5,29)(7,16)(8,17)(9,18)(10,19)(12,21)(13,22)(14,23)(15,24), (1,25)(5,29)(6,30)(10,19)(11,20)(15,24), (1,25)(4,28)(6,30)(9,18)(11,20)(14,23), (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)(16,18)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25) );
 
G=PermutationGroup([(3,27),(5,29),(8,17),(10,19),(13,22),(15,24)], [(2,26),(3,27),(4,28),(5,29),(7,16),(8,17),(9,18),(10,19),(12,21),(13,22),(14,23),(15,24)], [(1,25),(5,29),(6,30),(10,19),(11,20),(15,24)], [(1,25),(4,28),(6,30),(9,18),(11,20),(14,23)], [(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),(16,18),(19,30),(20,29),(21,28),(22,27),(23,26),(24,25)])
 
G:=TransitiveGroup(30,119);
 

Matrix representation ►G ⊆ GL7(𝔽61)

1000000
0100000
0010000
00060000
00006000
0000010
0000001
,
1000000
0100000
0010000
00060000
00006000
00000600
00000060
,
1000000
0100000
00600000
0001000
00006000
0000010
0000001
,
1000000
0100000
00600000
0001000
0000100
0000010
00000060
,
231400000
394500000
0000100
0000010
0000001
0010000
0001000
,
60100000
0100000
0000100
0001000
0010000
0000001
0000010

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] >;
 

Character table of C24⋊D15

 class 12A2B2C2D34A4B4C5A5B6A6B6C15A15B15C15D
 size 1555602606060323210101032323232
ρ1111111111111111111    trivial
ρ21111-11-1-1-1111111111    linear of order 2
ρ322220-100022-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ4222202000-1+√5/2-1-√5/2222-1+√5/2-1-√5/2-1-√5/2-1+√5/2    orthogonal lifted from D5
ρ5222202000-1-√5/2-1+√5/2222-1-√5/2-1+√5/2-1+√5/2-1-√5/2    orthogonal lifted from D5
ρ622220-1000-1+√5/2-1-√5/2-1-1-1-ζ32ζ54+ζ32ζ5-ζ54ζ3ζ53-ζ3ζ52-ζ52-ζ3ζ53+ζ3ζ52-ζ53-ζ3ζ54+ζ3ζ5-ζ54    orthogonal lifted from D15
ρ722220-1000-1-√5/2-1+√5/2-1-1-1-ζ3ζ53+ζ3ζ52-ζ53-ζ32ζ54+ζ32ζ5-ζ54-ζ3ζ54+ζ3ζ5-ζ54ζ3ζ53-ζ3ζ52-ζ52    orthogonal lifted from D15
ρ822220-1000-1-√5/2-1+√5/2-1-1-1ζ3ζ53-ζ3ζ52-ζ52-ζ3ζ54+ζ3ζ5-ζ54-ζ32ζ54+ζ32ζ5-ζ54-ζ3ζ53+ζ3ζ52-ζ53    orthogonal lifted from D15
ρ922220-1000-1+√5/2-1-√5/2-1-1-1-ζ3ζ54+ζ3ζ5-ζ54-ζ3ζ53+ζ3ζ52-ζ53ζ3ζ53-ζ3ζ52-ζ52-ζ32ζ54+ζ32ζ5-ζ54    orthogonal lifted from D15
ρ1051-31-15-1110011-30000    orthogonal lifted from C24⋊D5
ρ1151-31151-1-10011-30000    orthogonal lifted from C24⋊D5
ρ12511-315-1-11001-310000    orthogonal lifted from C24⋊D5
ρ13511-3-1511-1001-310000    orthogonal lifted from C24⋊D5
ρ145-31115-11-100-3110000    orthogonal lifted from C24⋊D5
ρ155-311-151-1100-3110000    orthogonal lifted from C24⋊D5
ρ161022-60-500000-13-10000    orthogonal faithful
ρ1710-6220-5000003-1-10000    orthogonal faithful
ρ18102-620-500000-1-130000    orthogonal faithful

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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