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## G = C3×C24⋊C5order 240 = 24·3·5

### Direct product of C3 and C24⋊C5

Aliases: C3×C24⋊C5, C242C15, (C23×C6)⋊C5, SmallGroup(240,199)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×C24⋊C5
G = < a,b,c,d,e,f | a3=b2=c2=d2=e2=f5=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=bcd, cd=dc, ce=ec, fcf-1=cde, fdf-1=de=ed, fef-1=b >

Character table of C3×C24⋊C5

 class 1 2A 2B 2C 3A 3B 5A 5B 5C 5D 6A 6B 6C 6D 6E 6F 15A 15B 15C 15D 15E 15F 15G 15H size 1 5 5 5 1 1 16 16 16 16 5 5 5 5 5 5 16 16 16 16 16 16 16 16 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 trivial ρ2 1 1 1 1 ζ3 ζ32 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 linear of order 3 ρ3 1 1 1 1 ζ32 ζ3 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 linear of order 3 ρ4 1 1 1 1 1 1 ζ54 ζ52 ζ53 ζ5 1 1 1 1 1 1 ζ52 ζ5 ζ54 ζ52 ζ53 ζ5 ζ54 ζ53 linear of order 5 ρ5 1 1 1 1 1 1 ζ5 ζ53 ζ52 ζ54 1 1 1 1 1 1 ζ53 ζ54 ζ5 ζ53 ζ52 ζ54 ζ5 ζ52 linear of order 5 ρ6 1 1 1 1 1 1 ζ53 ζ54 ζ5 ζ52 1 1 1 1 1 1 ζ54 ζ52 ζ53 ζ54 ζ5 ζ52 ζ53 ζ5 linear of order 5 ρ7 1 1 1 1 1 1 ζ52 ζ5 ζ54 ζ53 1 1 1 1 1 1 ζ5 ζ53 ζ52 ζ5 ζ54 ζ53 ζ52 ζ54 linear of order 5 ρ8 1 1 1 1 ζ32 ζ3 ζ5 ζ53 ζ52 ζ54 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3ζ53 ζ32ζ54 ζ32ζ5 ζ32ζ53 ζ3ζ52 ζ3ζ54 ζ3ζ5 ζ32ζ52 linear of order 15 ρ9 1 1 1 1 ζ32 ζ3 ζ53 ζ54 ζ5 ζ52 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3ζ54 ζ32ζ52 ζ32ζ53 ζ32ζ54 ζ3ζ5 ζ3ζ52 ζ3ζ53 ζ32ζ5 linear of order 15 ρ10 1 1 1 1 ζ32 ζ3 ζ54 ζ52 ζ53 ζ5 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3ζ52 ζ32ζ5 ζ32ζ54 ζ32ζ52 ζ3ζ53 ζ3ζ5 ζ3ζ54 ζ32ζ53 linear of order 15 ρ11 1 1 1 1 ζ3 ζ32 ζ5 ζ53 ζ52 ζ54 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32ζ53 ζ3ζ54 ζ3ζ5 ζ3ζ53 ζ32ζ52 ζ32ζ54 ζ32ζ5 ζ3ζ52 linear of order 15 ρ12 1 1 1 1 ζ3 ζ32 ζ52 ζ5 ζ54 ζ53 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32ζ5 ζ3ζ53 ζ3ζ52 ζ3ζ5 ζ32ζ54 ζ32ζ53 ζ32ζ52 ζ3ζ54 linear of order 15 ρ13 1 1 1 1 ζ32 ζ3 ζ52 ζ5 ζ54 ζ53 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3ζ5 ζ32ζ53 ζ32ζ52 ζ32ζ5 ζ3ζ54 ζ3ζ53 ζ3ζ52 ζ32ζ54 linear of order 15 ρ14 1 1 1 1 ζ3 ζ32 ζ53 ζ54 ζ5 ζ52 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32ζ54 ζ3ζ52 ζ3ζ53 ζ3ζ54 ζ32ζ5 ζ32ζ52 ζ32ζ53 ζ3ζ5 linear of order 15 ρ15 1 1 1 1 ζ3 ζ32 ζ54 ζ52 ζ53 ζ5 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32ζ52 ζ3ζ5 ζ3ζ54 ζ3ζ52 ζ32ζ53 ζ32ζ5 ζ32ζ54 ζ3ζ53 linear of order 15 ρ16 5 1 1 -3 5 5 0 0 0 0 -3 -3 1 1 1 1 0 0 0 0 0 0 0 0 orthogonal lifted from C24⋊C5 ρ17 5 1 -3 1 5 5 0 0 0 0 1 1 1 1 -3 -3 0 0 0 0 0 0 0 0 orthogonal lifted from C24⋊C5 ρ18 5 -3 1 1 5 5 0 0 0 0 1 1 -3 -3 1 1 0 0 0 0 0 0 0 0 orthogonal lifted from C24⋊C5 ρ19 5 1 1 -3 -5-5√-3/2 -5+5√-3/2 0 0 0 0 3-3√-3/2 3+3√-3/2 ζ32 ζ3 ζ3 ζ32 0 0 0 0 0 0 0 0 complex faithful ρ20 5 1 -3 1 -5+5√-3/2 -5-5√-3/2 0 0 0 0 ζ32 ζ3 ζ3 ζ32 3+3√-3/2 3-3√-3/2 0 0 0 0 0 0 0 0 complex faithful ρ21 5 -3 1 1 -5+5√-3/2 -5-5√-3/2 0 0 0 0 ζ32 ζ3 3-3√-3/2 3+3√-3/2 ζ32 ζ3 0 0 0 0 0 0 0 0 complex faithful ρ22 5 1 1 -3 -5+5√-3/2 -5-5√-3/2 0 0 0 0 3+3√-3/2 3-3√-3/2 ζ3 ζ32 ζ32 ζ3 0 0 0 0 0 0 0 0 complex faithful ρ23 5 1 -3 1 -5-5√-3/2 -5+5√-3/2 0 0 0 0 ζ3 ζ32 ζ32 ζ3 3-3√-3/2 3+3√-3/2 0 0 0 0 0 0 0 0 complex faithful ρ24 5 -3 1 1 -5-5√-3/2 -5+5√-3/2 0 0 0 0 ζ3 ζ32 3+3√-3/2 3-3√-3/2 ζ3 ζ32 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of C3×C24⋊C5
On 30 points - transitive group 30T52
Generators in S30
(1 27 17)(2 28 18)(3 29 19)(4 30 20)(5 26 16)(6 24 14)(7 25 15)(8 21 11)(9 22 12)(10 23 13)
(1 11)(3 13)(8 27)(10 29)(17 21)(19 23)
(1 11)(2 12)(4 14)(5 15)(6 30)(7 26)(8 27)(9 28)(16 25)(17 21)(18 22)(20 24)
(3 13)(4 14)(6 30)(10 29)(19 23)(20 24)
(2 12)(4 14)(6 30)(9 28)(18 22)(20 24)
(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)

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

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

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

G:=TransitiveGroup(30,52);

C3×C24⋊C5 is a maximal subgroup of   C24⋊D15

Matrix representation of C3×C24⋊C5 in GL5(𝔽31)

 5 0 0 0 0 0 5 0 0 0 0 0 5 0 0 0 0 0 5 0 0 0 0 0 5
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 4 20 22 30 0 26 10 8 0 30
,
 1 0 0 0 0 0 30 0 0 0 0 0 1 0 0 0 11 0 1 0 26 0 8 0 30
,
 30 0 0 0 0 0 30 0 0 0 0 0 1 0 0 0 0 22 30 0 0 0 8 0 30
,
 1 0 0 0 0 0 1 0 0 0 0 0 30 0 0 4 20 0 30 0 0 0 23 0 1
,
 0 1 0 0 0 0 0 1 0 0 4 20 22 29 0 0 0 0 9 1 0 0 0 23 0

G:=sub<GL(5,GF(31))| [5,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,0,5],[1,0,0,4,26,0,1,0,20,10,0,0,1,22,8,0,0,0,30,0,0,0,0,0,30],[1,0,0,0,26,0,30,0,11,0,0,0,1,0,8,0,0,0,1,0,0,0,0,0,30],[30,0,0,0,0,0,30,0,0,0,0,0,1,22,8,0,0,0,30,0,0,0,0,0,30],[1,0,0,4,0,0,1,0,20,0,0,0,30,0,23,0,0,0,30,0,0,0,0,0,1],[0,0,4,0,0,1,0,20,0,0,0,1,22,0,0,0,0,29,9,23,0,0,0,1,0] >;

C3×C24⋊C5 in GAP, Magma, Sage, TeX

C_3\times C_2^4\rtimes C_5
% in TeX

G:=Group("C3xC2^4:C5");
// GroupNames label

G:=SmallGroup(240,199);
// by ID

G=gap.SmallGroup(240,199);
# by ID

G:=PCGroup([6,-3,-5,-2,2,2,2,728,1089,1660,2711]);
// Polycyclic

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

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