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G = C3×C24⋊C6order 288 = 25·32

Direct product of C3 and C24⋊C6

direct product, metabelian, soluble, monomial

Aliases: C3×C24⋊C6, C22⋊A43C6, C231(C3×A4), C22≀C2⋊C32, C242(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

C1C24 — C3×C24⋊C6
C1C22C24C23×C6C3×C22⋊A4 — C3×C24⋊C6
C24 — C3×C24⋊C6
C1C3

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 [×4], C3, C3 [×3], C4, C22, C22 [×9], C6 [×7], C2×C4, D4 [×2], C23, C23 [×3], C32, C12, A4 [×9], C2×C6, C2×C6 [×9], C22⋊C4, C2×D4, C24, C3×C6, C2×C12, C3×D4 [×2], C2×A4 [×3], C22×C6, C22×C6 [×3], C22≀C2, C3×A4 [×3], C3×C22⋊C4, C6×D4, C22⋊A4 [×3], C23×C6, C6×A4, C24⋊C6 [×3], C3×C22≀C2, C3×C22⋊A4, C3×C24⋊C6
Quotients: C1, C2, C3 [×4], C6 [×4], 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 14 21)(8 15 22)(9 16 23)(10 17 24)(11 18 19)(12 13 20)
(1 10)(4 24)(6 17)(8 12)(13 15)(20 22)
(1 12)(4 20)(6 13)(8 10)(15 17)(22 24)
(1 10)(2 7)(3 21)(4 24)(5 14)(6 17)(8 12)(9 11)(13 15)(16 18)(19 23)(20 22)
(1 12)(2 9)(3 23)(4 20)(5 16)(6 13)(7 11)(8 10)(14 18)(15 17)(19 21)(22 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)

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

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

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

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

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