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## G = C3×C24⋊D5order 480 = 25·3·5

### Direct product of C3 and C24⋊D5

Aliases: C3×C24⋊D5, C24⋊C53C6, (C23×C6)⋊1D5, C242(C3×D5), (C3×C24⋊C5)⋊2C2, SmallGroup(480,1194)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C24⋊C5 — C3×C24⋊D5
 Chief series C1 — C24 — C24⋊C5 — C3×C24⋊C5 — C3×C24⋊D5
 Lower central C24⋊C5 — C3×C24⋊D5
 Upper central C1 — C3

Generators and relations for C3×C24⋊D5
G = < a,b,c,d,e,f,g | a3=b2=c2=d2=e2=f5=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, geg=bc=cb, bd=db, fcf-1=gcg=be=eb, fbf-1=e, bg=gb, cd=dc, ce=ec, de=ed, fdf-1=bce, gdg=cde, fef-1=bcde, gfg=f-1 >

Subgroups: 592 in 86 conjugacy classes, 8 normal (all characteristic)
C1, C2, C3, C4, C22, C5, C6, C2×C4, D4, C23, D5, C12, C2×C6, C15, C22⋊C4, C2×D4, C24, C2×C12, C3×D4, C22×C6, C3×D5, C22≀C2, C3×C22⋊C4, C6×D4, C23×C6, C24⋊C5, C3×C22≀C2, C24⋊D5, C3×C24⋊C5, C3×C24⋊D5
Quotients: C1, C2, C3, C6, D5, C3×D5, C24⋊D5, C3×C24⋊D5

Character table of C3×C24⋊D5

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 4C 5A 5B 6A 6B 6C 6D 6E 6F 6G 6H 12A 12B 12C 12D 12E 12F 15A 15B 15C 15D size 1 5 5 5 20 1 1 20 20 20 32 32 5 5 5 5 5 5 20 20 20 20 20 20 20 20 32 32 32 32 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 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 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ3 1 1 1 1 -1 ζ3 ζ32 -1 -1 -1 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ65 ζ6 ζ65 ζ6 ζ6 ζ6 ζ65 ζ65 ζ3 ζ3 ζ32 ζ32 linear of order 6 ρ4 1 1 1 1 -1 ζ32 ζ3 -1 -1 -1 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ6 ζ65 ζ6 ζ65 ζ65 ζ65 ζ6 ζ6 ζ32 ζ32 ζ3 ζ3 linear of order 6 ρ5 1 1 1 1 1 ζ3 ζ32 1 1 1 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ6 1 1 1 1 1 ζ32 ζ3 1 1 1 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ7 2 2 2 2 0 2 2 0 0 0 -1+√5/2 -1-√5/2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ8 2 2 2 2 0 2 2 0 0 0 -1-√5/2 -1+√5/2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ9 2 2 2 2 0 -1-√-3 -1+√-3 0 0 0 -1-√5/2 -1+√5/2 -1-√-3 -1+√-3 -1+√-3 -1+√-3 -1-√-3 -1-√-3 0 0 0 0 0 0 0 0 ζ32ζ53+ζ32ζ52 ζ32ζ54+ζ32ζ5 ζ3ζ54+ζ3ζ5 ζ3ζ53+ζ3ζ52 complex lifted from C3×D5 ρ10 2 2 2 2 0 -1+√-3 -1-√-3 0 0 0 -1+√5/2 -1-√5/2 -1+√-3 -1-√-3 -1-√-3 -1-√-3 -1+√-3 -1+√-3 0 0 0 0 0 0 0 0 ζ3ζ54+ζ3ζ5 ζ3ζ53+ζ3ζ52 ζ32ζ53+ζ32ζ52 ζ32ζ54+ζ32ζ5 complex lifted from C3×D5 ρ11 2 2 2 2 0 -1+√-3 -1-√-3 0 0 0 -1-√5/2 -1+√5/2 -1+√-3 -1-√-3 -1-√-3 -1-√-3 -1+√-3 -1+√-3 0 0 0 0 0 0 0 0 ζ3ζ53+ζ3ζ52 ζ3ζ54+ζ3ζ5 ζ32ζ54+ζ32ζ5 ζ32ζ53+ζ32ζ52 complex lifted from C3×D5 ρ12 2 2 2 2 0 -1-√-3 -1+√-3 0 0 0 -1+√5/2 -1-√5/2 -1-√-3 -1+√-3 -1+√-3 -1+√-3 -1-√-3 -1-√-3 0 0 0 0 0 0 0 0 ζ32ζ54+ζ32ζ5 ζ32ζ53+ζ32ζ52 ζ3ζ53+ζ3ζ52 ζ3ζ54+ζ3ζ5 complex lifted from C3×D5 ρ13 5 -3 1 1 -1 5 5 1 1 -1 0 0 -3 1 -3 1 1 1 -1 -1 1 -1 1 1 1 -1 0 0 0 0 orthogonal lifted from C24⋊D5 ρ14 5 1 1 -3 1 5 5 -1 1 -1 0 0 1 1 1 -3 -3 1 1 1 -1 -1 -1 1 1 -1 0 0 0 0 orthogonal lifted from C24⋊D5 ρ15 5 1 -3 1 1 5 5 1 -1 -1 0 0 1 -3 1 1 1 -3 1 1 1 -1 1 -1 -1 -1 0 0 0 0 orthogonal lifted from C24⋊D5 ρ16 5 1 -3 1 -1 5 5 -1 1 1 0 0 1 -3 1 1 1 -3 -1 -1 -1 1 -1 1 1 1 0 0 0 0 orthogonal lifted from C24⋊D5 ρ17 5 1 1 -3 -1 5 5 1 -1 1 0 0 1 1 1 -3 -3 1 -1 -1 1 1 1 -1 -1 1 0 0 0 0 orthogonal lifted from C24⋊D5 ρ18 5 -3 1 1 1 5 5 -1 -1 1 0 0 -3 1 -3 1 1 1 1 1 -1 1 -1 -1 -1 1 0 0 0 0 orthogonal lifted from C24⋊D5 ρ19 5 -3 1 1 -1 -5-5√-3/2 -5+5√-3/2 1 1 -1 0 0 3+3√-3/2 ζ3 3-3√-3/2 ζ3 ζ32 ζ32 ζ6 ζ65 ζ32 ζ65 ζ3 ζ3 ζ32 ζ6 0 0 0 0 complex faithful ρ20 5 1 -3 1 1 -5+5√-3/2 -5-5√-3/2 1 -1 -1 0 0 ζ3 3+3√-3/2 ζ32 ζ32 ζ3 3-3√-3/2 ζ3 ζ32 ζ3 ζ6 ζ32 ζ6 ζ65 ζ65 0 0 0 0 complex faithful ρ21 5 1 1 -3 1 -5-5√-3/2 -5+5√-3/2 -1 1 -1 0 0 ζ32 ζ3 ζ3 3-3√-3/2 3+3√-3/2 ζ32 ζ32 ζ3 ζ6 ζ65 ζ65 ζ3 ζ32 ζ6 0 0 0 0 complex faithful ρ22 5 1 -3 1 -1 -5+5√-3/2 -5-5√-3/2 -1 1 1 0 0 ζ3 3+3√-3/2 ζ32 ζ32 ζ3 3-3√-3/2 ζ65 ζ6 ζ65 ζ32 ζ6 ζ32 ζ3 ζ3 0 0 0 0 complex faithful ρ23 5 1 -3 1 1 -5-5√-3/2 -5+5√-3/2 1 -1 -1 0 0 ζ32 3-3√-3/2 ζ3 ζ3 ζ32 3+3√-3/2 ζ32 ζ3 ζ32 ζ65 ζ3 ζ65 ζ6 ζ6 0 0 0 0 complex faithful ρ24 5 1 1 -3 1 -5+5√-3/2 -5-5√-3/2 -1 1 -1 0 0 ζ3 ζ32 ζ32 3+3√-3/2 3-3√-3/2 ζ3 ζ3 ζ32 ζ65 ζ6 ζ6 ζ32 ζ3 ζ65 0 0 0 0 complex faithful ρ25 5 1 1 -3 -1 -5+5√-3/2 -5-5√-3/2 1 -1 1 0 0 ζ3 ζ32 ζ32 3+3√-3/2 3-3√-3/2 ζ3 ζ65 ζ6 ζ3 ζ32 ζ32 ζ6 ζ65 ζ3 0 0 0 0 complex faithful ρ26 5 -3 1 1 1 -5-5√-3/2 -5+5√-3/2 -1 -1 1 0 0 3+3√-3/2 ζ3 3-3√-3/2 ζ3 ζ32 ζ32 ζ32 ζ3 ζ6 ζ3 ζ65 ζ65 ζ6 ζ32 0 0 0 0 complex faithful ρ27 5 1 1 -3 -1 -5-5√-3/2 -5+5√-3/2 1 -1 1 0 0 ζ32 ζ3 ζ3 3-3√-3/2 3+3√-3/2 ζ32 ζ6 ζ65 ζ32 ζ3 ζ3 ζ65 ζ6 ζ32 0 0 0 0 complex faithful ρ28 5 1 -3 1 -1 -5-5√-3/2 -5+5√-3/2 -1 1 1 0 0 ζ32 3-3√-3/2 ζ3 ζ3 ζ32 3+3√-3/2 ζ6 ζ65 ζ6 ζ3 ζ65 ζ3 ζ32 ζ32 0 0 0 0 complex faithful ρ29 5 -3 1 1 -1 -5+5√-3/2 -5-5√-3/2 1 1 -1 0 0 3-3√-3/2 ζ32 3+3√-3/2 ζ32 ζ3 ζ3 ζ65 ζ6 ζ3 ζ6 ζ32 ζ32 ζ3 ζ65 0 0 0 0 complex faithful ρ30 5 -3 1 1 1 -5+5√-3/2 -5-5√-3/2 -1 -1 1 0 0 3-3√-3/2 ζ32 3+3√-3/2 ζ32 ζ3 ζ3 ζ3 ζ32 ζ65 ζ32 ζ6 ζ6 ζ65 ζ3 0 0 0 0 complex faithful

Permutation representations of C3×C24⋊D5
On 30 points - transitive group 30T113
Generators in S30
(1 28 18)(2 29 19)(3 30 20)(4 26 16)(5 27 17)(6 21 11)(7 22 12)(8 23 13)(9 24 14)(10 25 15)
(1 13)(2 14)(4 11)(5 12)(6 26)(7 27)(8 28)(9 29)(16 21)(17 22)(18 23)(19 24)
(3 15)(4 11)(6 26)(10 30)(16 21)(20 25)
(3 15)(5 12)(7 27)(10 30)(17 22)(20 25)
(1 13)(3 15)(4 11)(5 12)(6 26)(7 27)(8 28)(10 30)(16 21)(17 22)(18 23)(20 25)
(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 12)(2 11)(3 15)(4 14)(5 13)(6 29)(7 28)(8 27)(9 26)(10 30)(16 24)(17 23)(18 22)(19 21)(20 25)

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

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

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

G:=TransitiveGroup(30,113);

On 30 points - transitive group 30T118
Generators in S30
(1 28 18)(2 29 19)(3 30 20)(4 26 16)(5 27 17)(6 24 14)(7 25 15)(8 21 11)(9 22 12)(10 23 13)
(1 13)(2 14)(4 11)(5 12)(6 29)(8 26)(9 27)(10 28)(16 21)(17 22)(18 23)(19 24)
(3 15)(4 11)(7 30)(8 26)(16 21)(20 25)
(3 15)(5 12)(7 30)(9 27)(17 22)(20 25)
(1 13)(3 15)(4 11)(5 12)(7 30)(8 26)(9 27)(10 28)(16 21)(17 22)(18 23)(20 25)
(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 5)(2 4)(6 8)(9 10)(11 14)(12 13)(16 19)(17 18)(21 24)(22 23)(26 29)(27 28)

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

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

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

G:=TransitiveGroup(30,118);

Matrix representation of C3×C24⋊D5 in GL5(𝔽61)

 13 0 0 0 0 0 13 0 0 0 0 0 13 0 0 0 0 0 13 0 0 0 0 0 13
,
 60 0 0 0 0 0 60 0 0 0 0 0 1 0 0 0 0 0 60 0 0 0 0 0 60
,
 1 0 0 0 0 0 1 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 60 0 0 0 0 0 1 0 0 0 0 0 60
,
 60 0 0 0 0 0 1 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 0 0 60
,
 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0
,
 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0

G:=sub<GL(5,GF(61))| [13,0,0,0,0,0,13,0,0,0,0,0,13,0,0,0,0,0,13,0,0,0,0,0,13],[60,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,60],[60,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,1,0,0,0,0],[0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0] >;

C3×C24⋊D5 in GAP, Magma, Sage, TeX

C_3\times C_2^4\rtimes D_5
% in TeX

G:=Group("C3xC2^4:D5");
// GroupNames label

G:=SmallGroup(480,1194);
// by ID

G=gap.SmallGroup(480,1194);
# by ID

G:=PCGroup([7,-2,-3,-5,-2,2,2,2,506,2523,437,1068,13865,2539,7356,265]);
// Polycyclic

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

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