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## G = Dic3.A4order 144 = 24·32

### The non-split extension by Dic3 of A4 acting through Inn(Dic3)

Aliases: Dic3.A4, SL2(𝔽3)⋊2S3, C3⋊(C4.A4), (C3×Q8).C6, C2.2(S3×A4), C6.1(C2×A4), Q83S3⋊C3, Q8.2(C3×S3), (C3×SL2(𝔽3))⋊2C2, SmallGroup(144,127)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×Q8 — Dic3.A4
 Chief series C1 — C2 — C6 — C3×Q8 — C3×SL2(𝔽3) — Dic3.A4
 Lower central C3×Q8 — Dic3.A4
 Upper central C1 — C2

Generators and relations for Dic3.A4
G = < a,b,c,d,e | a6=e3=1, b2=c2=d2=a3, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=a3c, ece-1=a3cd, ede-1=c >

Character table of Dic3.A4

 class 1 2A 2B 3A 3B 3C 3D 3E 4A 4B 4C 6A 6B 6C 6D 6E 12A 12B 12C 12D 12E size 1 1 18 2 4 4 8 8 3 3 6 2 4 4 8 8 12 12 12 12 12 ρ1 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 linear of order 2 ρ3 1 1 1 1 ζ3 ζ32 ζ3 ζ32 1 1 1 1 ζ3 ζ32 ζ3 ζ32 1 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ4 1 1 1 1 ζ32 ζ3 ζ32 ζ3 1 1 1 1 ζ32 ζ3 ζ32 ζ3 1 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ5 1 1 -1 1 ζ32 ζ3 ζ32 ζ3 -1 -1 1 1 ζ32 ζ3 ζ32 ζ3 1 ζ65 ζ65 ζ6 ζ6 linear of order 6 ρ6 1 1 -1 1 ζ3 ζ32 ζ3 ζ32 -1 -1 1 1 ζ3 ζ32 ζ3 ζ32 1 ζ6 ζ6 ζ65 ζ65 linear of order 6 ρ7 2 2 0 -1 2 2 -1 -1 0 0 2 -1 2 2 -1 -1 -1 0 0 0 0 orthogonal lifted from S3 ρ8 2 2 0 -1 -1+√-3 -1-√-3 ζ65 ζ6 0 0 2 -1 -1+√-3 -1-√-3 ζ65 ζ6 -1 0 0 0 0 complex lifted from C3×S3 ρ9 2 2 0 -1 -1-√-3 -1+√-3 ζ6 ζ65 0 0 2 -1 -1-√-3 -1+√-3 ζ6 ζ65 -1 0 0 0 0 complex lifted from C3×S3 ρ10 2 -2 0 2 -1 -1 -1 -1 -2i 2i 0 -2 1 1 1 1 0 -i i -i i complex lifted from C4.A4 ρ11 2 -2 0 2 -1 -1 -1 -1 2i -2i 0 -2 1 1 1 1 0 i -i i -i complex lifted from C4.A4 ρ12 2 -2 0 2 ζ65 ζ6 ζ65 ζ6 2i -2i 0 -2 ζ3 ζ32 ζ3 ζ32 0 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ43ζ3 complex lifted from C4.A4 ρ13 2 -2 0 2 ζ6 ζ65 ζ6 ζ65 -2i 2i 0 -2 ζ32 ζ3 ζ32 ζ3 0 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ4ζ32 complex lifted from C4.A4 ρ14 2 -2 0 2 ζ6 ζ65 ζ6 ζ65 2i -2i 0 -2 ζ32 ζ3 ζ32 ζ3 0 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ43ζ32 complex lifted from C4.A4 ρ15 2 -2 0 2 ζ65 ζ6 ζ65 ζ6 -2i 2i 0 -2 ζ3 ζ32 ζ3 ζ32 0 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ4ζ3 complex lifted from C4.A4 ρ16 3 3 -1 3 0 0 0 0 3 3 -1 3 0 0 0 0 -1 0 0 0 0 orthogonal lifted from A4 ρ17 3 3 1 3 0 0 0 0 -3 -3 -1 3 0 0 0 0 -1 0 0 0 0 orthogonal lifted from C2×A4 ρ18 4 -4 0 -2 -2 -2 1 1 0 0 0 2 2 2 -1 -1 0 0 0 0 0 orthogonal faithful, Schur index 2 ρ19 4 -4 0 -2 1-√-3 1+√-3 ζ3 ζ32 0 0 0 2 -1+√-3 -1-√-3 ζ65 ζ6 0 0 0 0 0 complex faithful ρ20 4 -4 0 -2 1+√-3 1-√-3 ζ32 ζ3 0 0 0 2 -1-√-3 -1+√-3 ζ6 ζ65 0 0 0 0 0 complex faithful ρ21 6 6 0 -3 0 0 0 0 0 0 -2 -3 0 0 0 0 1 0 0 0 0 orthogonal lifted from S3×A4

Smallest permutation representation of Dic3.A4
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 28 4 25)(2 27 5 30)(3 26 6 29)(7 32 10 35)(8 31 11 34)(9 36 12 33)(13 38 16 41)(14 37 17 40)(15 42 18 39)(19 44 22 47)(20 43 23 46)(21 48 24 45)
(1 8 4 11)(2 9 5 12)(3 10 6 7)(13 19 16 22)(14 20 17 23)(15 21 18 24)(25 34 28 31)(26 35 29 32)(27 36 30 33)(37 43 40 46)(38 44 41 47)(39 45 42 48)
(1 14 4 17)(2 15 5 18)(3 16 6 13)(7 22 10 19)(8 23 11 20)(9 24 12 21)(25 40 28 37)(26 41 29 38)(27 42 30 39)(31 46 34 43)(32 47 35 44)(33 48 36 45)
(1 3 5)(2 4 6)(7 15 23)(8 16 24)(9 17 19)(10 18 20)(11 13 21)(12 14 22)(25 29 27)(26 30 28)(31 41 45)(32 42 46)(33 37 47)(34 38 48)(35 39 43)(36 40 44)

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

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

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

Dic3.A4 is a maximal subgroup of
CSU2(𝔽3)⋊S3  Dic3.4S4  Dic3.5S4  GL2(𝔽3)⋊S3  SL2(𝔽3).11D6  Dic6.A4  S3×C4.A4  Dic9.A4  Dic9.2A4  C6.(S3×A4)  C3⋊Dic3.2A4
Dic3.A4 is a maximal quotient of
Dic3×SL2(𝔽3)  Dic9.A4  Dic9.2A4  Q8⋊C93S3  C6.(S3×A4)  C3⋊Dic3.2A4

Matrix representation of Dic3.A4 in GL4(𝔽5) generated by

 0 2 2 1 4 1 4 3 4 2 0 2 3 1 4 1
,
 0 1 3 0 0 0 4 1 3 0 2 3 3 4 2 3
,
 2 4 3 1 1 0 4 2 2 2 2 3 0 1 2 1
,
 4 4 4 4 1 2 4 4 1 1 0 3 0 3 2 4
,
 4 0 1 4 0 4 2 0 0 2 0 0 1 2 0 0
G:=sub<GL(4,GF(5))| [0,4,4,3,2,1,2,1,2,4,0,4,1,3,2,1],[0,0,3,3,1,0,0,4,3,4,2,2,0,1,3,3],[2,1,2,0,4,0,2,1,3,4,2,2,1,2,3,1],[4,1,1,0,4,2,1,3,4,4,0,2,4,4,3,4],[4,0,0,1,0,4,2,2,1,2,0,0,4,0,0,0] >;

Dic3.A4 in GAP, Magma, Sage, TeX

{\rm Dic}_3.A_4
% in TeX

G:=Group("Dic3.A4");
// GroupNames label

G:=SmallGroup(144,127);
// by ID

G=gap.SmallGroup(144,127);
# by ID

G:=PCGroup([6,-2,-3,-2,2,-3,-2,432,170,230,81,351,165,1444]);
// Polycyclic

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

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