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## G = Dic3.5S4order 288 = 25·32

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

Aliases: Dic3.5S4, CSU2(𝔽3)⋊3S3, SL2(𝔽3).3D6, Q8.3S32, C2.6(S3×S4), C6.3(C2×S4), (C3×Q8).3D6, C6.6S41C2, Q83S32S3, C31(C4.6S4), Dic3.A42C2, (C3×CSU2(𝔽3))⋊1C2, (C3×SL2(𝔽3)).3C22, SmallGroup(288,846)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C3×SL2(𝔽3) — Dic3.5S4
 Chief series C1 — C2 — Q8 — C3×Q8 — C3×SL2(𝔽3) — Dic3.A4 — Dic3.5S4
 Lower central C3×SL2(𝔽3) — Dic3.5S4
 Upper central C1 — C2

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, C2×C4, D4, Q8, Q8, C32, Dic3, Dic3, C12, D6, C2×C8, D8, SD16, Q16, C4○D4, C3⋊S3, C3×C6, C3⋊C8, C24, SL2(𝔽3), SL2(𝔽3), C4×S3, D12, C3×Q8, C3×Q8, C4○D8, C3×Dic3, C2×C3⋊S3, S3×C8, D24, Q82S3, C3×Q16, CSU2(𝔽3), GL2(𝔽3), C4.A4, Q83S3, Q83S3, C6.D6, C3×SL2(𝔽3), D24⋊C2, C4.6S4, C3×CSU2(𝔽3), C6.6S4, Dic3.A4, Dic3.5S4
Quotients: C1, C2, C22, S3, D6, S4, S32, C2×S4, C4.6S4, S3×S4, Dic3.5S4

Character table of Dic3.5S4

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D 24A 24B size 1 1 18 36 2 8 16 3 3 6 12 2 8 16 6 6 18 18 12 24 24 24 12 12 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 1 1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 1 -1 -1 1 1 1 linear of order 2 ρ3 1 1 -1 1 1 1 1 -1 -1 1 -1 1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 -1 1 1 1 1 1 1 -1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 -1 linear of order 2 ρ5 2 2 2 0 2 -1 -1 2 2 2 0 2 -1 -1 0 0 0 0 2 -1 -1 0 0 0 orthogonal lifted from S3 ρ6 2 2 0 0 -1 2 -1 0 0 2 2 -1 2 -1 2 2 0 0 -1 0 0 -1 -1 -1 orthogonal lifted from S3 ρ7 2 2 -2 0 2 -1 -1 -2 -2 2 0 2 -1 -1 0 0 0 0 2 1 1 0 0 0 orthogonal lifted from D6 ρ8 2 2 0 0 -1 2 -1 0 0 2 -2 -1 2 -1 -2 -2 0 0 -1 0 0 1 1 1 orthogonal lifted from D6 ρ9 2 -2 0 0 2 -1 -1 2i -2i 0 0 -2 1 1 -√2 √2 √-2 -√-2 0 i -i 0 -√2 √2 complex lifted from C4.6S4 ρ10 2 -2 0 0 2 -1 -1 -2i 2i 0 0 -2 1 1 √2 -√2 √-2 -√-2 0 -i i 0 √2 -√2 complex lifted from C4.6S4 ρ11 2 -2 0 0 2 -1 -1 -2i 2i 0 0 -2 1 1 -√2 √2 -√-2 √-2 0 -i i 0 -√2 √2 complex lifted from C4.6S4 ρ12 2 -2 0 0 2 -1 -1 2i -2i 0 0 -2 1 1 √2 -√2 -√-2 √-2 0 i -i 0 √2 -√2 complex lifted from C4.6S4 ρ13 3 3 -1 1 3 0 0 3 3 -1 1 3 0 0 -1 -1 -1 -1 -1 0 0 1 -1 -1 orthogonal lifted from S4 ρ14 3 3 1 -1 3 0 0 -3 -3 -1 1 3 0 0 -1 -1 1 1 -1 0 0 1 -1 -1 orthogonal lifted from C2×S4 ρ15 3 3 1 1 3 0 0 -3 -3 -1 -1 3 0 0 1 1 -1 -1 -1 0 0 -1 1 1 orthogonal lifted from C2×S4 ρ16 3 3 -1 -1 3 0 0 3 3 -1 -1 3 0 0 1 1 1 1 -1 0 0 -1 1 1 orthogonal lifted from S4 ρ17 4 4 0 0 -2 -2 1 0 0 4 0 -2 -2 1 0 0 0 0 -2 0 0 0 0 0 orthogonal lifted from S32 ρ18 4 -4 0 0 -2 -2 1 0 0 0 0 2 2 -1 -2√2 2√2 0 0 0 0 0 0 √2 -√2 orthogonal faithful ρ19 4 -4 0 0 -2 -2 1 0 0 0 0 2 2 -1 2√2 -2√2 0 0 0 0 0 0 -√2 √2 orthogonal faithful ρ20 4 -4 0 0 4 1 1 -4i 4i 0 0 -4 -1 -1 0 0 0 0 0 i -i 0 0 0 complex lifted from C4.6S4 ρ21 4 -4 0 0 4 1 1 4i -4i 0 0 -4 -1 -1 0 0 0 0 0 -i i 0 0 0 complex lifted from C4.6S4 ρ22 6 6 0 0 -3 0 0 0 0 -2 -2 -3 0 0 2 2 0 0 1 0 0 1 -1 -1 orthogonal lifted from S3×S4 ρ23 6 6 0 0 -3 0 0 0 0 -2 2 -3 0 0 -2 -2 0 0 1 0 0 -1 1 1 orthogonal lifted from S3×S4 ρ24 8 -8 0 0 -4 2 -1 0 0 0 0 4 -2 1 0 0 0 0 0 0 0 0 0 0 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

 6 6 3 6 1 4 3 3 4 3 3 6 0 0 0 3
,
 4 0 3 3 6 0 2 4 3 4 1 1 3 3 2 2
,
 3 5 2 5 6 0 0 3 2 2 0 3 1 6 3 4
,
 1 5 4 0 2 0 1 5 4 4 3 6 3 4 2 3
,
 4 5 5 3 6 3 2 2 0 0 4 0 4 3 5 1
,
 4 1 1 4 3 0 3 2 2 2 3 3 5 2 1 0
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

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