Aliases: Dic3.4S4, GL2(𝔽3)⋊3S3, SL2(𝔽3).2D6, Q8.2S32, C2.5(S3×S4), C6.2(C2×S4), (C3×Q8).2D6, C6.5S4⋊2C2, Q8⋊3S3⋊1S3, C3⋊2(C4.6S4), Dic3.A4⋊1C2, (C3×GL2(𝔽3))⋊3C2, (C3×SL2(𝔽3)).2C22, SmallGroup(288,845)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C3×SL2(𝔽3) — Dic3.4S4 |
| C3×SL2(𝔽3) — Dic3.4S4 |
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, D4⋊2S3, Q8⋊3S3, 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 |
►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
Export