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G = C3×C22⋊S4order 288 = 25·32

Direct product of C3 and C22⋊S4

direct product, non-abelian, soluble, monomial

Aliases: C3×C22⋊S4, (C2×C6)⋊1S4, C22⋊(C3×S4), C22⋊A44C6, (C23×C6)⋊3S3, C245(C3×S3), (C3×C22⋊A4)⋊2C2, SmallGroup(288,1035)

Series: Derived Chief Lower central Upper central

C1C24C22⋊A4 — C3×C22⋊S4
C1C22C24C22⋊A4C3×C22⋊A4 — C3×C22⋊S4
C22⋊A4 — C3×C22⋊S4
C1C3

Generators and relations for C3×C22⋊S4
 G = < a,b,c,d,e,f,g | a3=b2=c2=d2=e2=f3=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, fcf-1=bc=cb, bd=db, be=eb, fbf-1=gbg=c, cd=dc, ce=ec, gcg=b, fdf-1=gdg=de=ed, fef-1=d, eg=ge, gfg=f-1 >

Subgroups: 574 in 119 conjugacy classes, 14 normal (8 characteristic)
C1, C2 [×5], C3, C3 [×2], C4 [×3], C22 [×3], C22 [×11], S3, C6 [×5], C2×C4 [×3], D4 [×6], C23 [×5], C32, C12 [×3], A4 [×9], C2×C6 [×3], C2×C6 [×11], C22⋊C4 [×3], C2×D4 [×3], C24, C3×S3, C2×C12 [×3], C3×D4 [×6], S4 [×3], C22×C6 [×5], C22≀C2, C3×A4 [×4], C3×C22⋊C4 [×3], C6×D4 [×3], C22⋊A4, C22⋊A4, C23×C6, C3×S4 [×3], C3×C22≀C2, C22⋊S4, C3×C22⋊A4, C3×C22⋊S4
Quotients: C1, C2, C3, S3, C6, C3×S3, S4 [×3], C3×S4 [×3], C22⋊S4, C3×C22⋊S4

Character table of C3×C22⋊S4

 class 12A2B2C2D2E3A3B3C3D3E4A4B4C6A6B6C6D6E6F6G6H6I6J12A12B12C12D12E12F
 size 133361211323232121212333333661212121212121212
ρ1111111111111111111111111111111    trivial
ρ211111-111111-1-1-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ311111-1ζ32ζ3ζ32ζ31-1-1-1ζ3ζ32ζ3ζ3ζ32ζ32ζ32ζ3ζ65ζ6ζ65ζ65ζ65ζ6ζ6ζ6    linear of order 6
ρ4111111ζ32ζ3ζ32ζ31111ζ3ζ32ζ3ζ3ζ32ζ32ζ32ζ3ζ3ζ32ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 3
ρ511111-1ζ3ζ32ζ3ζ321-1-1-1ζ32ζ3ζ32ζ32ζ3ζ3ζ3ζ32ζ6ζ65ζ6ζ6ζ6ζ65ζ65ζ65    linear of order 6
ρ6111111ζ3ζ32ζ3ζ321111ζ32ζ3ζ32ζ32ζ3ζ3ζ3ζ32ζ32ζ3ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 3
ρ722222022-1-1-10002222222200000000    orthogonal lifted from S3
ρ8222220-1--3-1+-3ζ6ζ65-1000-1+-3-1--3-1+-3-1+-3-1--3-1--3-1--3-1+-300000000    complex lifted from C3×S3
ρ9222220-1+-3-1--3ζ65ζ6-1000-1--3-1+-3-1--3-1--3-1+-3-1+-3-1+-3-1--300000000    complex lifted from C3×S3
ρ103-1-13-1-1330001-11-1-13-13-1-1-1-1-111-1-111    orthogonal lifted from S4
ρ113-13-1-1133000-1-11-1-1-13-13-1-1111-1-1-11-1    orthogonal lifted from S4
ρ1233-1-1-1-133000-11133-1-1-1-1-1-1-1-11-1111-1    orthogonal lifted from S4
ρ133-1-13-1133000-11-1-1-13-13-1-1-111-1-111-1-1    orthogonal lifted from S4
ρ143-13-1-1-13300011-1-1-1-13-13-1-1-1-1-1111-11    orthogonal lifted from S4
ρ1533-1-1-11330001-1-133-1-1-1-1-1-111-11-1-1-11    orthogonal lifted from S4
ρ163-1-13-11-3+3-3/2-3-3-3/2000-11-1ζ6ζ65-3-3-3/2ζ6-3+3-3/2ζ65ζ65ζ6ζ32ζ3ζ6ζ6ζ32ζ3ζ65ζ65    complex lifted from C3×S4
ρ173-13-1-11-3-3-3/2-3+3-3/2000-1-11ζ65ζ6ζ65-3+3-3/2ζ6-3-3-3/2ζ6ζ65ζ3ζ32ζ3ζ65ζ65ζ6ζ32ζ6    complex lifted from C3×S4
ρ1833-1-1-1-1-3+3-3/2-3-3-3/2000-111-3-3-3/2-3+3-3/2ζ6ζ6ζ65ζ65ζ65ζ6ζ6ζ65ζ32ζ6ζ32ζ3ζ3ζ65    complex lifted from C3×S4
ρ1933-1-1-1-1-3-3-3/2-3+3-3/2000-111-3+3-3/2-3-3-3/2ζ65ζ65ζ6ζ6ζ6ζ65ζ65ζ6ζ3ζ65ζ3ζ32ζ32ζ6    complex lifted from C3×S4
ρ2033-1-1-11-3-3-3/2-3+3-3/20001-1-1-3+3-3/2-3-3-3/2ζ65ζ65ζ6ζ6ζ6ζ65ζ3ζ32ζ65ζ3ζ65ζ6ζ6ζ32    complex lifted from C3×S4
ρ2133-1-1-11-3+3-3/2-3-3-3/20001-1-1-3-3-3/2-3+3-3/2ζ6ζ6ζ65ζ65ζ65ζ6ζ32ζ3ζ6ζ32ζ6ζ65ζ65ζ3    complex lifted from C3×S4
ρ223-1-13-1-1-3+3-3/2-3-3-3/20001-11ζ6ζ65-3-3-3/2ζ6-3+3-3/2ζ65ζ65ζ6ζ6ζ65ζ32ζ32ζ6ζ65ζ3ζ3    complex lifted from C3×S4
ρ233-13-1-1-1-3-3-3/2-3+3-3/200011-1ζ65ζ6ζ65-3+3-3/2ζ6-3-3-3/2ζ6ζ65ζ65ζ6ζ65ζ3ζ3ζ32ζ6ζ32    complex lifted from C3×S4
ρ243-13-1-11-3+3-3/2-3-3-3/2000-1-11ζ6ζ65ζ6-3-3-3/2ζ65-3+3-3/2ζ65ζ6ζ32ζ3ζ32ζ6ζ6ζ65ζ3ζ65    complex lifted from C3×S4
ρ253-13-1-1-1-3+3-3/2-3-3-3/200011-1ζ6ζ65ζ6-3-3-3/2ζ65-3+3-3/2ζ65ζ6ζ6ζ65ζ6ζ32ζ32ζ3ζ65ζ3    complex lifted from C3×S4
ρ263-1-13-11-3-3-3/2-3+3-3/2000-11-1ζ65ζ6-3+3-3/2ζ65-3-3-3/2ζ6ζ6ζ65ζ3ζ32ζ65ζ65ζ3ζ32ζ6ζ6    complex lifted from C3×S4
ρ273-1-13-1-1-3-3-3/2-3+3-3/20001-11ζ65ζ6-3+3-3/2ζ65-3-3-3/2ζ6ζ6ζ65ζ65ζ6ζ3ζ3ζ65ζ6ζ32ζ32    complex lifted from C3×S4
ρ286-2-2-22066000000-2-2-2-2-2-22200000000    orthogonal lifted from C22⋊S4
ρ296-2-2-220-3-3-3-3+3-30000001--31+-31--31--31+-31+-3-1--3-1+-300000000    complex faithful
ρ306-2-2-220-3+3-3-3-3-30000001+-31--31+-31+-31--31--3-1+-3-1--300000000    complex faithful

Permutation representations of C3×C22⋊S4
On 24 points - transitive group 24T700
Generators in S24
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(1 11)(2 12)(3 10)(4 24)(5 22)(6 23)(7 14)(8 15)(9 13)(16 21)(17 19)(18 20)
(1 18)(2 16)(3 17)(4 15)(5 13)(6 14)(7 23)(8 24)(9 22)(10 19)(11 20)(12 21)
(1 20)(2 21)(3 19)(4 15)(5 13)(6 14)(7 23)(8 24)(9 22)(10 17)(11 18)(12 16)
(1 18)(2 16)(3 17)(4 24)(5 22)(6 23)(7 14)(8 15)(9 13)(10 19)(11 20)(12 21)
(1 3 2)(4 22 14)(5 23 15)(6 24 13)(7 8 9)(10 21 18)(11 19 16)(12 20 17)
(1 7)(2 8)(3 9)(4 21)(5 19)(6 20)(10 22)(11 23)(12 24)(13 17)(14 18)(15 16)

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

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

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

G:=TransitiveGroup(24,700);

On 24 points - transitive group 24T701
Generators in S24
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(1 18)(2 16)(3 17)(4 20)(5 21)(6 19)(7 24)(8 22)(9 23)(10 15)(11 13)(12 14)
(1 15)(2 13)(3 14)(4 8)(5 9)(6 7)(10 18)(11 16)(12 17)(19 24)(20 22)(21 23)
(1 10)(2 11)(3 12)(4 8)(5 9)(6 7)(13 16)(14 17)(15 18)(19 24)(20 22)(21 23)
(1 15)(2 13)(3 14)(4 20)(5 21)(6 19)(7 24)(8 22)(9 23)(10 18)(11 16)(12 17)
(7 19 24)(8 20 22)(9 21 23)(10 15 18)(11 13 16)(12 14 17)
(1 4)(2 5)(3 6)(7 17)(8 18)(9 16)(10 22)(11 23)(12 24)(13 21)(14 19)(15 20)

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

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

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

G:=TransitiveGroup(24,701);

Matrix representation of C3×C22⋊S4 in GL6(𝔽13)

300000
030000
003000
000100
000010
000001
,
0112000
1012000
0012000
000001
000121212
000100
,
0121000
0120000
1120000
000010
000100
000121212
,
100000
010000
001000
000001
000121212
000100
,
100000
010000
001000
000121212
000001
000010
,
1120000
0121000
0120000
000100
000121212
000010
,
100000
001000
010000
000100
000001
000010

G:=sub<GL(6,GF(13))| [3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,12,12,12,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,1,12,0],[0,0,1,0,0,0,12,12,12,0,0,0,1,0,0,0,0,0,0,0,0,0,1,12,0,0,0,1,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,1,12,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12,0,1,0,0,0,12,1,0],[1,0,0,0,0,0,12,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12,1,0,0,0,0,12,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C3×C22⋊S4 in GAP, Magma, Sage, TeX

C_3\times C_2^2\rtimes S_4
% in TeX

G:=Group("C3xC2^2:S4");
// GroupNames label

G:=SmallGroup(288,1035);
// by ID

G=gap.SmallGroup(288,1035);
# by ID

G:=PCGroup([7,-2,-3,-3,-2,2,-2,2,254,1011,185,634,333,6053,1531,3534,608]);
// Polycyclic

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

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Character table of C3×C22⋊S4 in TeX

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