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G = Dic3.5S4order 288 = 25·32

2nd non-split extension by Dic3 of S4 acting through Inn(Dic3)

non-abelian, soluble

Aliases: Dic3.5S4, CSU2(F3):3S3, SL2(F3).3D6, Q8.3S32, C2.6(S3xS4), C6.3(C2xS4), (C3xQ8).3D6, C6.6S4:1C2, Q8:3S3:2S3, C3:1(C4.6S4), Dic3.A4:2C2, (C3xCSU2(F3)):1C2, (C3xSL2(F3)).3C22, SmallGroup(288,846)

Series: Derived Chief Lower central Upper central

C1C2Q8C3xSL2(F3) — Dic3.5S4
C1C2Q8C3xQ8C3xSL2(F3)Dic3.A4 — Dic3.5S4
C3xSL2(F3) — Dic3.5S4
C1C2

Generators and relations for Dic3.5S4
 G = < a,b,c,d,e,f | a6=e3=1, b2=c2=d2=f2=a3, bab-1=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd-1=a3c, ece-1=a3cd, fcf-1=cd, ede-1=c, fdf-1=a3d, fef-1=e-1 >

Subgroups: 542 in 85 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, C2xC4, D4, Q8, Q8, C32, Dic3, Dic3, C12, D6, C2xC8, D8, SD16, Q16, C4oD4, C3:S3, C3xC6, C3:C8, C24, SL2(F3), SL2(F3), C4xS3, D12, C3xQ8, C3xQ8, C4oD8, C3xDic3, C2xC3:S3, S3xC8, D24, Q8:2S3, C3xQ16, CSU2(F3), GL2(F3), C4.A4, Q8:3S3, Q8:3S3, C6.D6, C3xSL2(F3), D24:C2, C4.6S4, C3xCSU2(F3), C6.6S4, Dic3.A4, Dic3.5S4
Quotients: C1, C2, C22, S3, D6, S4, S32, C2xS4, C4.6S4, S3xS4, Dic3.5S4

Character table of Dic3.5S4

 class 12A2B2C3A3B3C4A4B4C4D6A6B6C8A8B8C8D12A12B12C12D24A24B
 size 1118362816336122816661818122424241212
ρ1111111111111111111111111    trivial
ρ211-1-1111-1-11111111-1-11-1-1111    linear of order 2
ρ311-11111-1-11-1111-1-1111-1-1-1-1-1    linear of order 2
ρ4111-1111111-1111-1-1-1-1111-1-1-1    linear of order 2
ρ522202-1-122202-1-100002-1-1000    orthogonal lifted from S3
ρ62200-12-10022-12-12200-100-1-1-1    orthogonal lifted from S3
ρ722-202-1-1-2-2202-1-10000211000    orthogonal lifted from D6
ρ82200-12-1002-2-12-1-2-200-100111    orthogonal lifted from D6
ρ92-2002-1-12i-2i00-211-22-2--20i-i0-22    complex lifted from C4.6S4
ρ102-2002-1-1-2i2i00-2112-2-2--20-ii02-2    complex lifted from C4.6S4
ρ112-2002-1-1-2i2i00-211-22--2-20-ii0-22    complex lifted from C4.6S4
ρ122-2002-1-12i-2i00-2112-2--2-20i-i02-2    complex lifted from C4.6S4
ρ1333-1130033-11300-1-1-1-1-1001-1-1    orthogonal lifted from S4
ρ14331-1300-3-3-11300-1-111-1001-1-1    orthogonal lifted from C2xS4
ρ153311300-3-3-1-130011-1-1-100-111    orthogonal lifted from C2xS4
ρ1633-1-130033-1-13001111-100-111    orthogonal lifted from S4
ρ174400-2-210040-2-210000-200000    orthogonal lifted from S32
ρ184-400-2-21000022-1-22220000002-2    orthogonal faithful
ρ194-400-2-21000022-122-22000000-22    orthogonal faithful
ρ204-400411-4i4i00-4-1-100000i-i000    complex lifted from C4.6S4
ρ214-4004114i-4i00-4-1-100000-ii000    complex lifted from C4.6S4
ρ226600-30000-2-2-30022001001-1-1    orthogonal lifted from S3xS4
ρ236600-30000-22-300-2-200100-111    orthogonal lifted from S3xS4
ρ248-800-42-100004-210000000000    orthogonal faithful

Smallest permutation representation of Dic3.5S4
On 48 points
Generators in S48
(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)(31 32 33 34 35 36)(37 38 39 40 41 42)(43 44 45 46 47 48)
(1 8 4 11)(2 7 5 10)(3 12 6 9)(13 35 16 32)(14 34 17 31)(15 33 18 36)(19 42 22 39)(20 41 23 38)(21 40 24 37)(25 46 28 43)(26 45 29 48)(27 44 30 47)
(1 25 4 28)(2 26 5 29)(3 27 6 30)(7 45 10 48)(8 46 11 43)(9 47 12 44)(13 41 16 38)(14 42 17 39)(15 37 18 40)(19 34 22 31)(20 35 23 32)(21 36 24 33)
(1 31 4 34)(2 32 5 35)(3 33 6 36)(7 13 10 16)(8 14 11 17)(9 15 12 18)(19 25 22 28)(20 26 23 29)(21 27 24 30)(37 47 40 44)(38 48 41 45)(39 43 42 46)
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 37 43)(14 38 44)(15 39 45)(16 40 46)(17 41 47)(18 42 48)(19 29 33)(20 30 34)(21 25 35)(22 26 36)(23 27 31)(24 28 32)
(1 11 4 8)(2 12 5 9)(3 7 6 10)(13 33 16 36)(14 34 17 31)(15 35 18 32)(19 46 22 43)(20 47 23 44)(21 48 24 45)(25 42 28 39)(26 37 29 40)(27 38 30 41)

G:=sub<Sym(48)| (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)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,8,4,11)(2,7,5,10)(3,12,6,9)(13,35,16,32)(14,34,17,31)(15,33,18,36)(19,42,22,39)(20,41,23,38)(21,40,24,37)(25,46,28,43)(26,45,29,48)(27,44,30,47), (1,25,4,28)(2,26,5,29)(3,27,6,30)(7,45,10,48)(8,46,11,43)(9,47,12,44)(13,41,16,38)(14,42,17,39)(15,37,18,40)(19,34,22,31)(20,35,23,32)(21,36,24,33), (1,31,4,34)(2,32,5,35)(3,33,6,36)(7,13,10,16)(8,14,11,17)(9,15,12,18)(19,25,22,28)(20,26,23,29)(21,27,24,30)(37,47,40,44)(38,48,41,45)(39,43,42,46), (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,37,43)(14,38,44)(15,39,45)(16,40,46)(17,41,47)(18,42,48)(19,29,33)(20,30,34)(21,25,35)(22,26,36)(23,27,31)(24,28,32), (1,11,4,8)(2,12,5,9)(3,7,6,10)(13,33,16,36)(14,34,17,31)(15,35,18,32)(19,46,22,43)(20,47,23,44)(21,48,24,45)(25,42,28,39)(26,37,29,40)(27,38,30,41)>;

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)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,8,4,11)(2,7,5,10)(3,12,6,9)(13,35,16,32)(14,34,17,31)(15,33,18,36)(19,42,22,39)(20,41,23,38)(21,40,24,37)(25,46,28,43)(26,45,29,48)(27,44,30,47), (1,25,4,28)(2,26,5,29)(3,27,6,30)(7,45,10,48)(8,46,11,43)(9,47,12,44)(13,41,16,38)(14,42,17,39)(15,37,18,40)(19,34,22,31)(20,35,23,32)(21,36,24,33), (1,31,4,34)(2,32,5,35)(3,33,6,36)(7,13,10,16)(8,14,11,17)(9,15,12,18)(19,25,22,28)(20,26,23,29)(21,27,24,30)(37,47,40,44)(38,48,41,45)(39,43,42,46), (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,37,43)(14,38,44)(15,39,45)(16,40,46)(17,41,47)(18,42,48)(19,29,33)(20,30,34)(21,25,35)(22,26,36)(23,27,31)(24,28,32), (1,11,4,8)(2,12,5,9)(3,7,6,10)(13,33,16,36)(14,34,17,31)(15,35,18,32)(19,46,22,43)(20,47,23,44)(21,48,24,45)(25,42,28,39)(26,37,29,40)(27,38,30,41) );

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),(25,26,27,28,29,30),(31,32,33,34,35,36),(37,38,39,40,41,42),(43,44,45,46,47,48)], [(1,8,4,11),(2,7,5,10),(3,12,6,9),(13,35,16,32),(14,34,17,31),(15,33,18,36),(19,42,22,39),(20,41,23,38),(21,40,24,37),(25,46,28,43),(26,45,29,48),(27,44,30,47)], [(1,25,4,28),(2,26,5,29),(3,27,6,30),(7,45,10,48),(8,46,11,43),(9,47,12,44),(13,41,16,38),(14,42,17,39),(15,37,18,40),(19,34,22,31),(20,35,23,32),(21,36,24,33)], [(1,31,4,34),(2,32,5,35),(3,33,6,36),(7,13,10,16),(8,14,11,17),(9,15,12,18),(19,25,22,28),(20,26,23,29),(21,27,24,30),(37,47,40,44),(38,48,41,45),(39,43,42,46)], [(1,5,3),(2,6,4),(7,9,11),(8,10,12),(13,37,43),(14,38,44),(15,39,45),(16,40,46),(17,41,47),(18,42,48),(19,29,33),(20,30,34),(21,25,35),(22,26,36),(23,27,31),(24,28,32)], [(1,11,4,8),(2,12,5,9),(3,7,6,10),(13,33,16,36),(14,34,17,31),(15,35,18,32),(19,46,22,43),(20,47,23,44),(21,48,24,45),(25,42,28,39),(26,37,29,40),(27,38,30,41)]])

Matrix representation of Dic3.5S4 in GL4(F7) generated by

6636
1433
4336
0003
,
4033
6024
3411
3322
,
3525
6003
2203
1634
,
1540
2015
4436
3423
,
4553
6322
0040
4351
,
4114
3032
2233
5210
G:=sub<GL(4,GF(7))| [6,1,4,0,6,4,3,0,3,3,3,0,6,3,6,3],[4,6,3,3,0,0,4,3,3,2,1,2,3,4,1,2],[3,6,2,1,5,0,2,6,2,0,0,3,5,3,3,4],[1,2,4,3,5,0,4,4,4,1,3,2,0,5,6,3],[4,6,0,4,5,3,0,3,5,2,4,5,3,2,0,1],[4,3,2,5,1,0,2,2,1,3,3,1,4,2,3,0] >;

Dic3.5S4 in GAP, Magma, Sage, TeX

{\rm Dic}_3._5S_4
% in TeX

G:=Group("Dic3.5S4");
// GroupNames label

G:=SmallGroup(288,846);
// by ID

G=gap.SmallGroup(288,846);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,1008,1016,93,675,1271,1908,172,768,1153,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^6=e^3=1,b^2=c^2=d^2=f^2=a^3,b*a*b^-1=a^-1,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,b*f=f*b,d*c*d^-1=a^3*c,e*c*e^-1=a^3*c*d,f*c*f^-1=c*d,e*d*e^-1=c,f*d*f^-1=a^3*d,f*e*f^-1=e^-1>;
// generators/relations

Export

Character table of Dic3.5S4 in TeX

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