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## G = Dic3.4S4order 288 = 25·32

### 1st non-split extension by Dic3 of S4 acting through Inn(Dic3)

Aliases: Dic3.4S4, GL2(𝔽3)⋊3S3, SL2(𝔽3).2D6, Q8.2S32, C2.5(S3×S4), C6.2(C2×S4), (C3×Q8).2D6, C6.5S42C2, Q83S31S3, C32(C4.6S4), Dic3.A41C2, (C3×GL2(𝔽3))⋊3C2, (C3×SL2(𝔽3)).2C22, SmallGroup(288,845)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C3×SL2(𝔽3) — Dic3.4S4
 Chief series C1 — C2 — Q8 — C3×Q8 — C3×SL2(𝔽3) — Dic3.A4 — Dic3.4S4
 Lower central C3×SL2(𝔽3) — Dic3.4S4
 Upper central C1 — C2

Generators and relations for Dic3.4S4
G = < a,b,c,d,e,f | a6=e3=f2=1, b2=c2=d2=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=cd, ede-1=c, fdf=a3d, fef=e-1 >

Subgroups: 454 in 83 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×C6, C2×C8, D8, SD16, Q16, C4○D4, C3×S3, C3×C6, C3⋊C8, C24, SL2(𝔽3), SL2(𝔽3), Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C4○D8, C3×Dic3, C3⋊Dic3, S3×C6, S3×C8, C24⋊C2, D4⋊S3, C3⋊Q16, C3×SD16, CSU2(𝔽3), GL2(𝔽3), C4.A4, D42S3, Q83S3, S3×Dic3, C3×SL2(𝔽3), Q8.7D6, C4.6S4, C3×GL2(𝔽3), C6.5S4, Dic3.A4, Dic3.4S4
Quotients: C1, C2, C22, S3, D6, S4, S32, C2×S4, C4.6S4, S3×S4, Dic3.4S4

Character table of Dic3.4S4

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

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

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

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

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

Matrix representation of Dic3.4S4 in GL4(𝔽73) generated by

 72 1 0 0 72 0 0 0 0 0 72 0 0 0 0 72
,
 72 0 0 0 72 1 0 0 0 0 27 0 0 0 0 27
,
 1 0 0 0 0 1 0 0 0 0 32 21 0 0 52 41
,
 1 0 0 0 0 1 0 0 0 0 20 21 0 0 40 53
,
 1 0 0 0 0 1 0 0 0 0 0 72 0 0 1 72
,
 72 0 0 0 0 72 0 0 0 0 1 72 0 0 0 72
G:=sub<GL(4,GF(73))| [72,72,0,0,1,0,0,0,0,0,72,0,0,0,0,72],[72,72,0,0,0,1,0,0,0,0,27,0,0,0,0,27],[1,0,0,0,0,1,0,0,0,0,32,52,0,0,21,41],[1,0,0,0,0,1,0,0,0,0,20,40,0,0,21,53],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,72,72],[72,0,0,0,0,72,0,0,0,0,1,0,0,0,72,72] >;

Dic3.4S4 in GAP, Magma, Sage, TeX

{\rm Dic}_3._4S_4
% in TeX

G:=Group("Dic3.4S4");
// GroupNames label

G:=SmallGroup(288,845);
// by ID

G=gap.SmallGroup(288,845);
# by ID

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

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

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