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## G = A4⋊Dic6order 288 = 25·32

### The semidirect product of A4 and Dic6 acting via Dic6/C12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C6×A4 — A4⋊Dic6
 Chief series C1 — C22 — C2×C6 — C3×A4 — C6×A4 — C6.7S4 — A4⋊Dic6
 Lower central C3×A4 — C6×A4 — A4⋊Dic6
 Upper central C1 — C2 — C4

Generators and relations for A4⋊Dic6
G = < a,b,c,d,e | a2=b2=c3=d12=1, e2=d6, cac-1=eae-1=ab=ba, ad=da, cbc-1=a, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 540 in 108 conjugacy classes, 27 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, C6, C6, C2×C4, Q8, C23, C32, Dic3, C12, C12, A4, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C3×C6, Dic6, C2×Dic3, C2×C12, C2×A4, C22×C6, C22⋊Q8, C3⋊Dic3, C3×C12, C3×A4, Dic3⋊C4, C4⋊Dic3, C6.D4, A4⋊C4, C4×A4, C2×Dic6, C22×C12, C324Q8, C6×A4, C12.48D4, A4⋊Q8, C6.7S4, C12×A4, A4⋊Dic6
Quotients: C1, C2, C22, S3, Q8, D6, C3⋊S3, Dic6, S4, C2×C3⋊S3, C2×S4, C324Q8, C3⋊S4, A4⋊Q8, C2×C3⋊S4, A4⋊Dic6

Character table of A4⋊Dic6

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J size 1 1 3 3 2 8 8 8 2 6 36 36 36 36 2 6 6 8 8 8 2 2 6 6 8 8 8 8 8 8 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 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 -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 -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 1 1 1 1 1 1 linear of order 2 ρ5 2 2 2 2 -1 2 -1 -1 2 2 0 0 0 0 -1 -1 -1 -1 -1 2 -1 -1 -1 -1 -1 2 -1 2 -1 -1 orthogonal lifted from S3 ρ6 2 2 2 2 -1 -1 -1 2 -2 -2 0 0 0 0 -1 -1 -1 2 -1 -1 1 1 1 1 -2 1 1 1 -2 1 orthogonal lifted from D6 ρ7 2 2 2 2 -1 -1 2 -1 2 2 0 0 0 0 -1 -1 -1 -1 2 -1 -1 -1 -1 -1 -1 -1 2 -1 -1 2 orthogonal lifted from S3 ρ8 2 2 2 2 -1 -1 2 -1 -2 -2 0 0 0 0 -1 -1 -1 -1 2 -1 1 1 1 1 1 1 -2 1 1 -2 orthogonal lifted from D6 ρ9 2 2 2 2 -1 -1 -1 2 2 2 0 0 0 0 -1 -1 -1 2 -1 -1 -1 -1 -1 -1 2 -1 -1 -1 2 -1 orthogonal lifted from S3 ρ10 2 2 2 2 -1 2 -1 -1 -2 -2 0 0 0 0 -1 -1 -1 -1 -1 2 1 1 1 1 1 -2 1 -2 1 1 orthogonal lifted from D6 ρ11 2 2 2 2 2 -1 -1 -1 2 2 0 0 0 0 2 2 2 -1 -1 -1 2 2 2 2 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 2 2 2 2 -1 -1 -1 -2 -2 0 0 0 0 2 2 2 -1 -1 -1 -2 -2 -2 -2 1 1 1 1 1 1 orthogonal lifted from D6 ρ13 2 -2 -2 2 2 2 2 2 0 0 0 0 0 0 -2 -2 2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ14 2 -2 -2 2 -1 -1 -1 2 0 0 0 0 0 0 1 1 -1 -2 1 1 -√3 √3 √3 -√3 0 √3 √3 -√3 0 -√3 symplectic lifted from Dic6, Schur index 2 ρ15 2 -2 -2 2 2 -1 -1 -1 0 0 0 0 0 0 -2 -2 2 1 1 1 0 0 0 0 √3 √3 -√3 -√3 -√3 √3 symplectic lifted from Dic6, Schur index 2 ρ16 2 -2 -2 2 -1 2 -1 -1 0 0 0 0 0 0 1 1 -1 1 1 -2 -√3 √3 √3 -√3 -√3 0 -√3 0 √3 √3 symplectic lifted from Dic6, Schur index 2 ρ17 2 -2 -2 2 -1 -1 -1 2 0 0 0 0 0 0 1 1 -1 -2 1 1 √3 -√3 -√3 √3 0 -√3 -√3 √3 0 √3 symplectic lifted from Dic6, Schur index 2 ρ18 2 -2 -2 2 -1 -1 2 -1 0 0 0 0 0 0 1 1 -1 1 -2 1 √3 -√3 -√3 √3 -√3 √3 0 -√3 √3 0 symplectic lifted from Dic6, Schur index 2 ρ19 2 -2 -2 2 -1 2 -1 -1 0 0 0 0 0 0 1 1 -1 1 1 -2 √3 -√3 -√3 √3 √3 0 √3 0 -√3 -√3 symplectic lifted from Dic6, Schur index 2 ρ20 2 -2 -2 2 -1 -1 2 -1 0 0 0 0 0 0 1 1 -1 1 -2 1 -√3 √3 √3 -√3 √3 -√3 0 √3 -√3 0 symplectic lifted from Dic6, Schur index 2 ρ21 2 -2 -2 2 2 -1 -1 -1 0 0 0 0 0 0 -2 -2 2 1 1 1 0 0 0 0 -√3 -√3 √3 √3 √3 -√3 symplectic lifted from Dic6, Schur index 2 ρ22 3 3 -1 -1 3 0 0 0 -3 1 -1 -1 1 1 3 -1 -1 0 0 0 -3 -3 1 1 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ23 3 3 -1 -1 3 0 0 0 3 -1 1 -1 1 -1 3 -1 -1 0 0 0 3 3 -1 -1 0 0 0 0 0 0 orthogonal lifted from S4 ρ24 3 3 -1 -1 3 0 0 0 3 -1 -1 1 -1 1 3 -1 -1 0 0 0 3 3 -1 -1 0 0 0 0 0 0 orthogonal lifted from S4 ρ25 3 3 -1 -1 3 0 0 0 -3 1 1 1 -1 -1 3 -1 -1 0 0 0 -3 -3 1 1 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ26 6 6 -2 -2 -3 0 0 0 -6 2 0 0 0 0 -3 1 1 0 0 0 3 3 -1 -1 0 0 0 0 0 0 orthogonal lifted from C2×C3⋊S4 ρ27 6 6 -2 -2 -3 0 0 0 6 -2 0 0 0 0 -3 1 1 0 0 0 -3 -3 1 1 0 0 0 0 0 0 orthogonal lifted from C3⋊S4 ρ28 6 -6 2 -2 6 0 0 0 0 0 0 0 0 0 -6 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from A4⋊Q8, Schur index 2 ρ29 6 -6 2 -2 -3 0 0 0 0 0 0 0 0 0 3 -1 1 0 0 0 -3√3 3√3 -√3 √3 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ30 6 -6 2 -2 -3 0 0 0 0 0 0 0 0 0 3 -1 1 0 0 0 3√3 -3√3 √3 -√3 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of A4⋊Dic6
On 72 points
Generators in S72
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 13)(10 14)(11 15)(12 16)(25 47)(26 48)(27 37)(28 38)(29 39)(30 40)(31 41)(32 42)(33 43)(34 44)(35 45)(36 46)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)(33 37)(34 38)(35 39)(36 40)(49 69)(50 70)(51 71)(52 72)(53 61)(54 62)(55 63)(56 64)(57 65)(58 66)(59 67)(60 68)
(1 47 71)(2 48 72)(3 37 61)(4 38 62)(5 39 63)(6 40 64)(7 41 65)(8 42 66)(9 43 67)(10 44 68)(11 45 69)(12 46 70)(13 27 53)(14 28 54)(15 29 55)(16 30 56)(17 31 57)(18 32 58)(19 33 59)(20 34 60)(21 35 49)(22 36 50)(23 25 51)(24 26 52)
(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)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)
(1 14 7 20)(2 13 8 19)(3 24 9 18)(4 23 10 17)(5 22 11 16)(6 21 12 15)(25 68 31 62)(26 67 32 61)(27 66 33 72)(28 65 34 71)(29 64 35 70)(30 63 36 69)(37 52 43 58)(38 51 44 57)(39 50 45 56)(40 49 46 55)(41 60 47 54)(42 59 48 53)

G:=sub<Sym(72)| (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,13)(10,14)(11,15)(12,16)(25,47)(26,48)(27,37)(28,38)(29,39)(30,40)(31,41)(32,42)(33,43)(34,44)(35,45)(36,46)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(33,37)(34,38)(35,39)(36,40)(49,69)(50,70)(51,71)(52,72)(53,61)(54,62)(55,63)(56,64)(57,65)(58,66)(59,67)(60,68), (1,47,71)(2,48,72)(3,37,61)(4,38,62)(5,39,63)(6,40,64)(7,41,65)(8,42,66)(9,43,67)(10,44,68)(11,45,69)(12,46,70)(13,27,53)(14,28,54)(15,29,55)(16,30,56)(17,31,57)(18,32,58)(19,33,59)(20,34,60)(21,35,49)(22,36,50)(23,25,51)(24,26,52), (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,14,7,20)(2,13,8,19)(3,24,9,18)(4,23,10,17)(5,22,11,16)(6,21,12,15)(25,68,31,62)(26,67,32,61)(27,66,33,72)(28,65,34,71)(29,64,35,70)(30,63,36,69)(37,52,43,58)(38,51,44,57)(39,50,45,56)(40,49,46,55)(41,60,47,54)(42,59,48,53)>;

G:=Group( (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,13)(10,14)(11,15)(12,16)(25,47)(26,48)(27,37)(28,38)(29,39)(30,40)(31,41)(32,42)(33,43)(34,44)(35,45)(36,46)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(33,37)(34,38)(35,39)(36,40)(49,69)(50,70)(51,71)(52,72)(53,61)(54,62)(55,63)(56,64)(57,65)(58,66)(59,67)(60,68), (1,47,71)(2,48,72)(3,37,61)(4,38,62)(5,39,63)(6,40,64)(7,41,65)(8,42,66)(9,43,67)(10,44,68)(11,45,69)(12,46,70)(13,27,53)(14,28,54)(15,29,55)(16,30,56)(17,31,57)(18,32,58)(19,33,59)(20,34,60)(21,35,49)(22,36,50)(23,25,51)(24,26,52), (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,14,7,20)(2,13,8,19)(3,24,9,18)(4,23,10,17)(5,22,11,16)(6,21,12,15)(25,68,31,62)(26,67,32,61)(27,66,33,72)(28,65,34,71)(29,64,35,70)(30,63,36,69)(37,52,43,58)(38,51,44,57)(39,50,45,56)(40,49,46,55)(41,60,47,54)(42,59,48,53) );

G=PermutationGroup([[(1,17),(2,18),(3,19),(4,20),(5,21),(6,22),(7,23),(8,24),(9,13),(10,14),(11,15),(12,16),(25,47),(26,48),(27,37),(28,38),(29,39),(30,40),(31,41),(32,42),(33,43),(34,44),(35,45),(36,46),(49,55),(50,56),(51,57),(52,58),(53,59),(54,60),(61,67),(62,68),(63,69),(64,70),(65,71),(66,72)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,41),(26,42),(27,43),(28,44),(29,45),(30,46),(31,47),(32,48),(33,37),(34,38),(35,39),(36,40),(49,69),(50,70),(51,71),(52,72),(53,61),(54,62),(55,63),(56,64),(57,65),(58,66),(59,67),(60,68)], [(1,47,71),(2,48,72),(3,37,61),(4,38,62),(5,39,63),(6,40,64),(7,41,65),(8,42,66),(9,43,67),(10,44,68),(11,45,69),(12,46,70),(13,27,53),(14,28,54),(15,29,55),(16,30,56),(17,31,57),(18,32,58),(19,33,59),(20,34,60),(21,35,49),(22,36,50),(23,25,51),(24,26,52)], [(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),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72)], [(1,14,7,20),(2,13,8,19),(3,24,9,18),(4,23,10,17),(5,22,11,16),(6,21,12,15),(25,68,31,62),(26,67,32,61),(27,66,33,72),(28,65,34,71),(29,64,35,70),(30,63,36,69),(37,52,43,58),(38,51,44,57),(39,50,45,56),(40,49,46,55),(41,60,47,54),(42,59,48,53)]])

Matrix representation of A4⋊Dic6 in GL5(𝔽13)

 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 2 1 0 0 0 0 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 11 12 0 0 0 1 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 11 11 0 0 0 8 2 1 0 0 5 12 0
,
 2 9 0 0 0 9 2 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 3 6 0 0 0 7 10 0 0 0 0 0 1 0 0 0 0 5 0 1 0 0 8 1 0

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,12,2,0,0,0,0,1,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,11,1,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,11,8,5,0,0,11,2,12,0,0,0,1,0],[2,9,0,0,0,9,2,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[3,7,0,0,0,6,10,0,0,0,0,0,1,5,8,0,0,0,0,1,0,0,0,1,0] >;

A4⋊Dic6 in GAP, Magma, Sage, TeX

A_4\rtimes {\rm Dic}_6
% in TeX

G:=Group("A4:Dic6");
// GroupNames label

G:=SmallGroup(288,907);
// by ID

G=gap.SmallGroup(288,907);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,28,85,36,451,1684,6053,782,3534,1350]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^12=1,e^2=d^6,c*a*c^-1=e*a*e^-1=a*b=b*a,a*d=d*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations

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