Copied to
clipboard

G = C3×C24⋊C6  order 288 = 25·32

Direct product of C3 and C24⋊C6

direct product, metabelian, soluble, monomial

Aliases: C3×C24⋊C6, C22⋊A4⋊3C6, C23⋊1(C3×A4), C22≀C2⋊C32, C24⋊2(C3×C6), (C22×C6)⋊1A4, (C23×C6)⋊1C6, C22.2(C6×A4), (C3×C22≀C2)⋊C3, (C3×C22⋊A4)⋊1C2, (C2×C6).10(C2×A4), SmallGroup(288,634)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C24 — C3×C24⋊C6
Chief series
C1 — C22 — C24 — C23×C6 — C3×C22⋊A4 — C3×C24⋊C6
Lower central
C24 — C3×C24⋊C6
Upper central
C1 — C3

Generators and relations for C3×C24⋊C6
 G = < a,b,c,d,e,f | a3=b2=c2=d2=e2=f6=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=ec=ce, cd=dc, fcf-1=bcde, fef-1=de=ed, fdf-1=e >

Subgroups: 444 in 86 conjugacy classes, 18 normal (12 characteristic)
C1, C2, C3, C3, C4, C22, C22, C6, C2×C4, D4, C23, C23, C32, C12, A4, C2×C6, C2×C6, C22⋊C4, C2×D4, C24, C3×C6, C2×C12, C3×D4, C2×A4, C22×C6, C22×C6, C22≀C2, C3×A4, C3×C22⋊C4, C6×D4, C22⋊A4, C23×C6, C6×A4, C24⋊C6, C3×C22≀C2, C3×C22⋊A4, C3×C24⋊C6
Quotients: C1, C2, C3, C6, C32, A4, C3×C6, C2×A4, C3×A4, C6×A4, C24⋊C6, C3×C24⋊C6

Character table of C3×C24⋊C6

 class 12A2B2C2D3A3B3C3D3E3F3G3H46A6B6C6D6E6F6G6H6I6J6K6L6M6N12A12B
 size 134661116161616161612334466661616161616161212
ρ1111111111111111111111111111111    trivial
ρ211-11111111111-111-1-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ311-111ζ3ζ321ζ31ζ32ζ3ζ32-1ζ3ζ32ζ65ζ6ζ3ζ3ζ32ζ32ζ65ζ6ζ65-1ζ6-1ζ6ζ65    linear of order 6
ρ411-11111ζ32ζ3ζ3ζ32ζ32ζ3-111-1-11111ζ6ζ65ζ65ζ6ζ6ζ65-1-1    linear of order 6
ρ511111ζ3ζ32ζ31ζ321ζ32ζ31ζ3ζ32ζ3ζ32ζ3ζ3ζ32ζ32ζ32ζ31ζ31ζ32ζ32ζ3    linear of order 3
ρ611111ζ32ζ31ζ321ζ3ζ32ζ31ζ32ζ3ζ32ζ3ζ32ζ32ζ3ζ3ζ32ζ3ζ321ζ31ζ3ζ32    linear of order 3
ρ71111111ζ3ζ32ζ32ζ3ζ3ζ32111111111ζ3ζ32ζ32ζ3ζ3ζ3211    linear of order 3
ρ811111ζ32ζ3ζ321ζ31ζ3ζ321ζ32ζ3ζ32ζ3ζ32ζ32ζ3ζ3ζ3ζ321ζ321ζ3ζ3ζ32    linear of order 3
ρ911-111ζ32ζ3ζ3ζ3ζ32ζ3211-1ζ32ζ3ζ6ζ65ζ32ζ32ζ3ζ3-1-1ζ65ζ65ζ6ζ6ζ65ζ6    linear of order 6
ρ1011111ζ3ζ32ζ32ζ32ζ3ζ3111ζ3ζ32ζ3ζ32ζ3ζ3ζ32ζ3211ζ32ζ32ζ3ζ3ζ32ζ3    linear of order 3
ρ1111111ζ3ζ321ζ31ζ32ζ3ζ321ζ3ζ32ζ3ζ32ζ3ζ3ζ32ζ32ζ3ζ32ζ31ζ321ζ32ζ3    linear of order 3
ρ1211-111ζ32ζ31ζ321ζ3ζ32ζ3-1ζ32ζ3ζ6ζ65ζ32ζ32ζ3ζ3ζ6ζ65ζ6-1ζ65-1ζ65ζ6    linear of order 6
ρ1311-11111ζ3ζ32ζ32ζ3ζ3ζ32-111-1-11111ζ65ζ6ζ6ζ65ζ65ζ6-1-1    linear of order 6
ρ141111111ζ32ζ3ζ3ζ32ζ32ζ3111111111ζ32ζ3ζ3ζ32ζ32ζ311    linear of order 3
ρ1511111ζ32ζ3ζ3ζ3ζ32ζ32111ζ32ζ3ζ32ζ3ζ32ζ32ζ3ζ311ζ3ζ3ζ32ζ32ζ3ζ32    linear of order 3
ρ1611-111ζ3ζ32ζ31ζ321ζ32ζ3-1ζ3ζ32ζ65ζ6ζ3ζ3ζ32ζ32ζ6ζ65-1ζ65-1ζ6ζ6ζ65    linear of order 6
ρ1711-111ζ32ζ3ζ321ζ31ζ3ζ32-1ζ32ζ3ζ6ζ65ζ32ζ32ζ3ζ3ζ65ζ6-1ζ6-1ζ65ζ65ζ6    linear of order 6
ρ1811-111ζ3ζ32ζ32ζ32ζ3ζ311-1ζ3ζ32ζ65ζ6ζ3ζ3ζ32ζ32-1-1ζ6ζ6ζ65ζ65ζ6ζ65    linear of order 6
ρ19333-1-133000000-13333-1-1-1-1000000-1-1    orthogonal lifted from A4
ρ2033-3-1-133000000133-3-3-1-1-1-100000011    orthogonal lifted from C2×A4
ρ21333-1-1-3+3√-3/2-3-3√-3/2000000-1-3+3√-3/2-3-3√-3/2-3+3√-3/2-3-3√-3/2ζ65ζ65ζ6ζ6000000ζ6ζ65    complex lifted from C3×A4
ρ2233-3-1-1-3-3√-3/2-3+3√-3/20000001-3-3√-3/2-3+3√-3/23+3√-3/23-3√-3/2ζ6ζ6ζ65ζ65000000ζ3ζ32    complex lifted from C6×A4
ρ23333-1-1-3-3√-3/2-3+3√-3/2000000-1-3-3√-3/2-3+3√-3/2-3-3√-3/2-3+3√-3/2ζ6ζ6ζ65ζ65000000ζ65ζ6    complex lifted from C3×A4
ρ2433-3-1-1-3+3√-3/2-3-3√-3/20000001-3+3√-3/2-3-3√-3/23-3√-3/23+3√-3/2ζ65ζ65ζ6ζ6000000ζ32ζ3    complex lifted from C6×A4
ρ256-20-22660000000-2-200-22-2200000000    orthogonal lifted from C24⋊C6
ρ266-202-2660000000-2-2002-22-200000000    orthogonal lifted from C24⋊C6
ρ276-202-2-3-3√-3-3+3√-300000001+√-31-√-300-1-√-31+√-3-1+√-31-√-300000000    complex faithful
ρ286-202-2-3+3√-3-3-3√-300000001-√-31+√-300-1+√-31-√-3-1-√-31+√-300000000    complex faithful
ρ296-20-22-3-3√-3-3+3√-300000001+√-31-√-3001+√-3-1-√-31-√-3-1+√-300000000    complex faithful
ρ306-20-22-3+3√-3-3-3√-300000001-√-31+√-3001-√-3-1+√-31+√-3-1-√-300000000    complex faithful

Permutation representations of C3×C24⋊C6
►On 24 points - transitive group 24T698
Generators in S24
(1 6 4)(2 5 3)(7 21 13)(8 22 14)(9 23 15)(10 24 16)(11 19 17)(12 20 18)

(2 15)(3 23)(5 9)(7 11)(13 17)(19 21)

(2 17)(3 19)(5 11)(7 9)(13 15)(21 23)

(1 18)(2 15)(3 23)(4 20)(5 9)(6 12)(7 11)(8 10)(13 17)(14 16)(19 21)(22 24)

(1 14)(2 17)(3 19)(4 22)(5 11)(6 8)(7 9)(10 12)(13 15)(16 18)(20 24)(21 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)
 
G:=sub<Sym(24)| (1,6,4)(2,5,3)(7,21,13)(8,22,14)(9,23,15)(10,24,16)(11,19,17)(12,20,18), (2,15)(3,23)(5,9)(7,11)(13,17)(19,21), (2,17)(3,19)(5,11)(7,9)(13,15)(21,23), (1,18)(2,15)(3,23)(4,20)(5,9)(6,12)(7,11)(8,10)(13,17)(14,16)(19,21)(22,24), (1,14)(2,17)(3,19)(4,22)(5,11)(6,8)(7,9)(10,12)(13,15)(16,18)(20,24)(21,23), (1,2)(3,4)(5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)>;
 
G:=Group( (1,6,4)(2,5,3)(7,21,13)(8,22,14)(9,23,15)(10,24,16)(11,19,17)(12,20,18), (2,15)(3,23)(5,9)(7,11)(13,17)(19,21), (2,17)(3,19)(5,11)(7,9)(13,15)(21,23), (1,18)(2,15)(3,23)(4,20)(5,9)(6,12)(7,11)(8,10)(13,17)(14,16)(19,21)(22,24), (1,14)(2,17)(3,19)(4,22)(5,11)(6,8)(7,9)(10,12)(13,15)(16,18)(20,24)(21,23), (1,2)(3,4)(5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24) );
 
G=PermutationGroup([[(1,6,4),(2,5,3),(7,21,13),(8,22,14),(9,23,15),(10,24,16),(11,19,17),(12,20,18)], [(2,15),(3,23),(5,9),(7,11),(13,17),(19,21)], [(2,17),(3,19),(5,11),(7,9),(13,15),(21,23)], [(1,18),(2,15),(3,23),(4,20),(5,9),(6,12),(7,11),(8,10),(13,17),(14,16),(19,21),(22,24)], [(1,14),(2,17),(3,19),(4,22),(5,11),(6,8),(7,9),(10,12),(13,15),(16,18),(20,24),(21,23)], [(1,2),(3,4),(5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)]])
 
G:=TransitiveGroup(24,698);
 

Matrix representation of C3×C24⋊C6 ►in GL6(𝔽13)

900000
090000
009000
000900
000090
000009
,
0121000
0120000
1120000
0000112
0001012
0000012
,
0112000
1012000
0012000
0001200
0001201
0001210
,
1200000
1201000
1210000
0001200
0001201
0001210
,
0121000
0120000
1120000
0000121
0000120
0001120
,
000010
000001
000100
010000
001000
100000

G:=sub<GL(6,GF(13))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[0,0,1,0,0,0,12,12,12,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,12,12,12],[0,1,0,0,0,0,1,0,0,0,0,0,12,12,12,0,0,0,0,0,0,12,12,12,0,0,0,0,0,1,0,0,0,0,1,0],[12,12,12,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,12,12,0,0,0,0,0,1,0,0,0,0,1,0],[0,0,1,0,0,0,12,12,12,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,12,12,12,0,0,0,1,0,0],[0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0] >;
 

C3×C24⋊C6 in GAP, Magma, Sage, TeX 

C_3\times C_2^4\rtimes C_6
 
% in TeX
 
G:=Group("C3xC2^4:C6");
 
// GroupNames label
 
G:=SmallGroup(288,634);
 
// by ID
 
G=gap.SmallGroup(288,634);
 
# by ID
 
G:=PCGroup([7,-2,-3,-3,-2,2,-2,2,2523,514,6304,956,3036,5305]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=e^2=f^6=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=e*c=c*e,c*d=d*c,f*c*f^-1=b*c*d*e,f*e*f^-1=d*e=e*d,f*d*f^-1=e>;
 
// generators/relations
 

Export

Character table of C3×C24⋊C6 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁