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G = C24⋊4D15  order 480 = 25·3·5

3rd semidirect product of C24 and D15 acting via D15/C5=S3

non-abelian, soluble, monomial

Aliases: C24⋊4D15, C5⋊(C22⋊S4), C22⋊(C5⋊S4), (C2×C10)⋊2S4, C22⋊A4⋊3D5, (C23×C10)⋊6S3, (C5×C22⋊A4)⋊2C2, SmallGroup(480,1201)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C24 — C5×C22⋊A4 — C24⋊4D15
Chief series
C1 — C22 — C24 — C23×C10 — C5×C22⋊A4 — C24⋊4D15

Subgroups: 1084 in 112 conjugacy classes, 13 normal (7 characteristic)
C1, C2 [×5], C3, C4 [×3], C22 [×3], C22 [×11], C5, S3, C2×C4 [×3], D4 [×6], C23 [×5], D5, C10 [×4], A4 [×4], C15, C22⋊C4 [×3], C2×D4 [×3], C24, Dic5 [×3], D10 [×3], C2×C10 [×3], C2×C10 [×8], S4 [×3], D15, C22≀C2, C2×Dic5 [×3], C5⋊D4 [×6], C22×D5, C22×C10 [×4], C22⋊A4, C5×A4 [×4], C23.D5 [×3], C2×C5⋊D4 [×3], C23×C10, C22⋊S4, C5⋊S4 [×3], C24⋊2D5, C5×C22⋊A4, C24⋊4D15

Quotients:
C1, C2, S3, D5, S4 [×3], D15, C22⋊S4, C5⋊S4 [×3], C24⋊4D15

Generators and relations
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e15=f2=1, eae-1=fbf=ab=ba, ac=ca, ad=da, af=fa, bc=cb, bd=db, ebe-1=a, fcf=ede-1=cd=dc, ece-1=d, df=fd, fef=e-1 >

Smallest permutation representation
►On 40 points
Generators in S40
(1 38)(2 29)(3 35)(4 26)(5 32)(6 17)(7 23)(8 14)(9 20)(10 11)(12 22)(13 18)(15 25)(16 21)(19 24)(27 37)(28 33)(30 40)(31 36)(34 39)

(1 33)(2 39)(3 30)(4 36)(5 27)(6 12)(7 18)(8 24)(9 15)(10 21)(11 16)(13 23)(14 19)(17 22)(20 25)(26 31)(28 38)(29 34)(32 37)(35 40)

(1 38)(2 29)(3 35)(4 26)(5 32)(6 22)(7 13)(8 19)(9 25)(10 16)(11 21)(12 17)(14 24)(15 20)(18 23)(27 37)(28 33)(30 40)(31 36)(34 39)

(1 28)(2 34)(3 40)(4 31)(5 37)(6 12)(7 18)(8 24)(9 15)(10 21)(11 16)(13 23)(14 19)(17 22)(20 25)(26 36)(27 32)(29 39)(30 35)(33 38)

(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)

(1 6)(2 10)(3 9)(4 8)(5 7)(11 29)(12 28)(13 27)(14 26)(15 40)(16 39)(17 38)(18 37)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)(25 30)
 
G:=sub<Sym(40)| (1,38)(2,29)(3,35)(4,26)(5,32)(6,17)(7,23)(8,14)(9,20)(10,11)(12,22)(13,18)(15,25)(16,21)(19,24)(27,37)(28,33)(30,40)(31,36)(34,39), (1,33)(2,39)(3,30)(4,36)(5,27)(6,12)(7,18)(8,24)(9,15)(10,21)(11,16)(13,23)(14,19)(17,22)(20,25)(26,31)(28,38)(29,34)(32,37)(35,40), (1,38)(2,29)(3,35)(4,26)(5,32)(6,22)(7,13)(8,19)(9,25)(10,16)(11,21)(12,17)(14,24)(15,20)(18,23)(27,37)(28,33)(30,40)(31,36)(34,39), (1,28)(2,34)(3,40)(4,31)(5,37)(6,12)(7,18)(8,24)(9,15)(10,21)(11,16)(13,23)(14,19)(17,22)(20,25)(26,36)(27,32)(29,39)(30,35)(33,38), (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), (1,6)(2,10)(3,9)(4,8)(5,7)(11,29)(12,28)(13,27)(14,26)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)>;
 
G:=Group( (1,38)(2,29)(3,35)(4,26)(5,32)(6,17)(7,23)(8,14)(9,20)(10,11)(12,22)(13,18)(15,25)(16,21)(19,24)(27,37)(28,33)(30,40)(31,36)(34,39), (1,33)(2,39)(3,30)(4,36)(5,27)(6,12)(7,18)(8,24)(9,15)(10,21)(11,16)(13,23)(14,19)(17,22)(20,25)(26,31)(28,38)(29,34)(32,37)(35,40), (1,38)(2,29)(3,35)(4,26)(5,32)(6,22)(7,13)(8,19)(9,25)(10,16)(11,21)(12,17)(14,24)(15,20)(18,23)(27,37)(28,33)(30,40)(31,36)(34,39), (1,28)(2,34)(3,40)(4,31)(5,37)(6,12)(7,18)(8,24)(9,15)(10,21)(11,16)(13,23)(14,19)(17,22)(20,25)(26,36)(27,32)(29,39)(30,35)(33,38), (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), (1,6)(2,10)(3,9)(4,8)(5,7)(11,29)(12,28)(13,27)(14,26)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30) );
 
G=PermutationGroup([(1,38),(2,29),(3,35),(4,26),(5,32),(6,17),(7,23),(8,14),(9,20),(10,11),(12,22),(13,18),(15,25),(16,21),(19,24),(27,37),(28,33),(30,40),(31,36),(34,39)], [(1,33),(2,39),(3,30),(4,36),(5,27),(6,12),(7,18),(8,24),(9,15),(10,21),(11,16),(13,23),(14,19),(17,22),(20,25),(26,31),(28,38),(29,34),(32,37),(35,40)], [(1,38),(2,29),(3,35),(4,26),(5,32),(6,22),(7,13),(8,19),(9,25),(10,16),(11,21),(12,17),(14,24),(15,20),(18,23),(27,37),(28,33),(30,40),(31,36),(34,39)], [(1,28),(2,34),(3,40),(4,31),(5,37),(6,12),(7,18),(8,24),(9,15),(10,21),(11,16),(13,23),(14,19),(17,22),(20,25),(26,36),(27,32),(29,39),(30,35),(33,38)], [(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)], [(1,6),(2,10),(3,9),(4,8),(5,7),(11,29),(12,28),(13,27),(14,26),(15,40),(16,39),(17,38),(18,37),(19,36),(20,35),(21,34),(22,33),(23,32),(24,31),(25,30)])
 

Matrix representation ►G ⊆ GL8(𝔽61)

10000000
01000000
00100000
00010000
00001000
000006000
000006001
000006010
,
10000000
01000000
00100000
00010000
00001000
000000601
000000600
000001600
,
10000000
01000000
006000000
006001000
006010000
000000601
000000600
000001600
,
10000000
01000000
000601000
000600000
001600000
000006000
000006001
000006010
,
3012000000
4942000000
00010000
00001000
00100000
00000001
00000100
00000010
,
2427000000
5137000000
00001000
00010000
00100000
00000100
00000001
00000010

G:=sub<GL(8,GF(61))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,60,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,60,60,60,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,60,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,60,60,60,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,60,60,60,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,60,60,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[30,49,0,0,0,0,0,0,12,42,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[24,51,0,0,0,0,0,0,27,37,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
 

Character table of C24⋊4D15

 class 12A2B2C2D2E34A4B4C5A5B10A10B10C10D10E10F10G10H10I10J15A15B15C15D
 size 13336603260606022666666666632323232
ρ111111111111111111111111111    trivial
ρ211111-11-1-1-11111111111111111    linear of order 2
ρ3222220-1000222222222222-1-1-1-1    orthogonal lifted from S3
ρ42222202000-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2    orthogonal lifted from D5
ρ52222202000-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2    orthogonal lifted from D5
ρ6222220-1000-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1+√5/2-ζ32ζ54+ζ32ζ5-ζ54ζ3ζ53-ζ3ζ52-ζ52-ζ3ζ53+ζ3ζ52-ζ53-ζ3ζ54+ζ3ζ5-ζ54    orthogonal lifted from D15
ρ7222220-1000-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1-√5/2ζ3ζ53-ζ3ζ52-ζ52-ζ3ζ54+ζ3ζ5-ζ54-ζ32ζ54+ζ32ζ5-ζ54-ζ3ζ53+ζ3ζ52-ζ53    orthogonal lifted from D15
ρ8222220-1000-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1-√5/2-ζ3ζ53+ζ3ζ52-ζ53-ζ32ζ54+ζ32ζ5-ζ54-ζ3ζ54+ζ3ζ5-ζ54ζ3ζ53-ζ3ζ52-ζ52    orthogonal lifted from D15
ρ9222220-1000-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1+√5/2-ζ3ζ54+ζ3ζ5-ζ54-ζ3ζ53+ζ3ζ52-ζ53ζ3ζ53-ζ3ζ52-ζ52-ζ32ζ54+ζ32ζ5-ζ54    orthogonal lifted from D15
ρ1033-1-1-1-1011-133-1-1-1-1-133-1-1-10000    orthogonal lifted from S4
ρ113-13-1-1-101-1133-13-1-1-1-1-13-1-10000    orthogonal lifted from S4
ρ123-1-13-1-10-11133-1-1-13-1-1-1-1-130000    orthogonal lifted from S4
ρ1333-1-1-110-1-1133-1-1-1-1-133-1-1-10000    orthogonal lifted from S4
ρ143-13-1-110-11-133-13-1-1-1-1-13-1-10000    orthogonal lifted from S4
ρ153-1-13-1101-1-133-1-1-13-1-1-1-1-130000    orthogonal lifted from S4
ρ166-2-2-2200000662-22-22-2-2-22-20000    orthogonal lifted from C22⋊S4
ρ1766-2-2-200000-3-3√5/2-3+3√5/21+√5/21-√5/21-√5/21-√5/21-√5/2-3+3√5/2-3-3√5/21+√5/21+√5/21+√5/20000    orthogonal lifted from C5⋊S4
ρ1866-2-2-200000-3+3√5/2-3-3√5/21-√5/21+√5/21+√5/21+√5/21+√5/2-3-3√5/2-3+3√5/21-√5/21-√5/21-√5/20000    orthogonal lifted from C5⋊S4
ρ196-26-2-200000-3+3√5/2-3-3√5/21-√5/2-3-3√5/21+√5/21+√5/21+√5/21+√5/21-√5/2-3+3√5/21-√5/21-√5/20000    orthogonal lifted from C5⋊S4
ρ206-2-26-200000-3-3√5/2-3+3√5/21+√5/21-√5/21-√5/2-3+3√5/21-√5/21-√5/21+√5/21+√5/21+√5/2-3-3√5/20000    orthogonal lifted from C5⋊S4
ρ216-26-2-200000-3-3√5/2-3+3√5/21+√5/2-3+3√5/21-√5/21-√5/21-√5/21-√5/21+√5/2-3-3√5/21+√5/21+√5/20000    orthogonal lifted from C5⋊S4
ρ226-2-26-200000-3+3√5/2-3-3√5/21-√5/21+√5/21+√5/2-3-3√5/21+√5/21+√5/21-√5/21-√5/21-√5/2-3+3√5/20000    orthogonal lifted from C5⋊S4
ρ236-2-2-2200000-3-3√5/2-3+3√5/23ζ53-ζ521-√5/23ζ54-ζ51-√5/2-ζ54+3ζ51-√5/21+√5/21+√5/2-ζ53+3ζ521+√5/20000    complex faithful
ρ246-2-2-2200000-3+3√5/2-3-3√5/23ζ54-ζ51+√5/2-ζ53+3ζ521+√5/23ζ53-ζ521+√5/21-√5/21-√5/2-ζ54+3ζ51-√5/20000    complex faithful
ρ256-2-2-2200000-3+3√5/2-3-3√5/2-ζ54+3ζ51+√5/23ζ53-ζ521+√5/2-ζ53+3ζ521+√5/21-√5/21-√5/23ζ54-ζ51-√5/20000    complex faithful
ρ266-2-2-2200000-3-3√5/2-3+3√5/2-ζ53+3ζ521-√5/2-ζ54+3ζ51-√5/23ζ54-ζ51-√5/21+√5/21+√5/23ζ53-ζ521+√5/20000    complex faithful

In GAP, Magma, Sage, TeX 

C_2^4\rtimes_4D_{15}
 
% in TeX
 
G:=Group("C2^4:4D15");
 
// GroupNames label
 
G:=SmallGroup(480,1201);
 
// by ID
 
G=gap.SmallGroup(480,1201);
 
# by ID
 
G:=PCGroup([7,-2,-3,-5,-2,2,-2,2,57,506,1683,850,1054,1586,10085,7572,5886,2953]);
 
// 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=f*b*f=a*b=b*a,a*c=c*a,a*d=d*a,a*f=f*a,b*c=c*b,b*d=d*b,e*b*e^-1=a,f*c*f=e*d*e^-1=c*d=d*c,e*c*e^-1=d,d*f=f*d,f*e*f=e^-1>;
 
// generators/relations
 

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