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G = A4⋊Dic6order 288 = 25·32

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

non-abelian, soluble, monomial

Aliases: C12.5S4, A42Dic6, (C3×A4)⋊3Q8, C4.1(C3⋊S4), C6.28(C2×S4), (C4×A4).1S3, C32(A4⋊Q8), (C2×A4).8D6, (C2×C6)⋊3Dic6, (C12×A4).1C2, C6.7S4.2C2, (C22×C12).4S3, (C22×C6).19D6, C22⋊(C324Q8), (C6×A4).13C22, C2.3(C2×C3⋊S4), C23.1(C2×C3⋊S3), (C22×C4).2(C3⋊S3), SmallGroup(288,907)

Series: Derived Chief Lower central Upper central

C1C22C6×A4 — A4⋊Dic6
C1C22C2×C6C3×A4C6×A4C6.7S4 — A4⋊Dic6
C3×A4C6×A4 — A4⋊Dic6
C1C2C4

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 [×2], C3, C3 [×3], C4, C4 [×5], C22, C22 [×2], C6, C6 [×5], C2×C4 [×6], Q8 [×2], C23, C32, Dic3 [×10], C12, C12 [×4], A4 [×3], C2×C6, C2×C6 [×2], C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, C3×C6, Dic6 [×5], C2×Dic3 [×4], C2×C12 [×2], C2×A4 [×3], C22×C6, C22⋊Q8, C3⋊Dic3 [×2], C3×C12, C3×A4, Dic3⋊C4 [×2], C4⋊Dic3, C6.D4 [×2], A4⋊C4 [×6], C4×A4 [×3], C2×Dic6, C22×C12, C324Q8, C6×A4, C12.48D4, A4⋊Q8 [×3], C6.7S4 [×2], C12×A4, A4⋊Dic6
Quotients: C1, C2 [×3], C22, S3 [×4], Q8, D6 [×4], C3⋊S3, Dic6 [×4], S4, C2×C3⋊S3, C2×S4, C324Q8, C3⋊S4, A4⋊Q8, C2×C3⋊S4, A4⋊Dic6

Character table of A4⋊Dic6

 class 12A2B2C3A3B3C3D4A4B4C4D4E4F6A6B6C6D6E6F12A12B12C12D12E12F12G12H12I12J
 size 1133288826363636362668882266888888
ρ1111111111111111111111111111111    trivial
ρ211111111-1-11-1-11111111-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ311111111-1-1-111-1111111-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ41111111111-1-1-1-11111111111111111    linear of order 2
ρ52222-12-1-1220000-1-1-1-1-12-1-1-1-1-12-12-1-1    orthogonal lifted from S3
ρ62222-1-1-12-2-20000-1-1-12-1-11111-2111-21    orthogonal lifted from D6
ρ72222-1-12-1220000-1-1-1-12-1-1-1-1-1-1-12-1-12    orthogonal lifted from S3
ρ82222-1-12-1-2-20000-1-1-1-12-1111111-211-2    orthogonal lifted from D6
ρ92222-1-1-12220000-1-1-12-1-1-1-1-1-12-1-1-12-1    orthogonal lifted from S3
ρ102222-12-1-1-2-20000-1-1-1-1-1211111-21-211    orthogonal lifted from D6
ρ1122222-1-1-1220000222-1-1-12222-1-1-1-1-1-1    orthogonal lifted from S3
ρ1222222-1-1-1-2-20000222-1-1-1-2-2-2-2111111    orthogonal lifted from D6
ρ132-2-222222000000-2-22-2-2-20000000000    symplectic lifted from Q8, Schur index 2
ρ142-2-22-1-1-1200000011-1-211-333-3033-30-3    symplectic lifted from Dic6, Schur index 2
ρ152-2-222-1-1-1000000-2-22111000033-3-3-33    symplectic lifted from Dic6, Schur index 2
ρ162-2-22-12-1-100000011-111-2-333-3-30-3033    symplectic lifted from Dic6, Schur index 2
ρ172-2-22-1-1-1200000011-1-2113-3-330-3-3303    symplectic lifted from Dic6, Schur index 2
ρ182-2-22-1-12-100000011-11-213-3-33-330-330    symplectic lifted from Dic6, Schur index 2
ρ192-2-22-12-1-100000011-111-23-3-333030-3-3    symplectic lifted from Dic6, Schur index 2
ρ202-2-22-1-12-100000011-11-21-333-33-303-30    symplectic lifted from Dic6, Schur index 2
ρ212-2-222-1-1-1000000-2-221110000-3-3333-3    symplectic lifted from Dic6, Schur index 2
ρ2233-1-13000-31-1-1113-1-1000-3-311000000    orthogonal lifted from C2×S4
ρ2333-1-130003-11-11-13-1-100033-1-1000000    orthogonal lifted from S4
ρ2433-1-130003-1-11-113-1-100033-1-1000000    orthogonal lifted from S4
ρ2533-1-13000-3111-1-13-1-1000-3-311000000    orthogonal lifted from C2×S4
ρ2666-2-2-3000-620000-31100033-1-1000000    orthogonal lifted from C2×C3⋊S4
ρ2766-2-2-30006-20000-311000-3-311000000    orthogonal lifted from C3⋊S4
ρ286-62-26000000000-62-20000000000000    symplectic lifted from A4⋊Q8, Schur index 2
ρ296-62-2-30000000003-11000-3333-33000000    symplectic faithful, Schur index 2
ρ306-62-2-30000000003-1100033-333-3000000    symplectic faithful, Schur index 2

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

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

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

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

Matrix representation of A4⋊Dic6 in GL5(𝔽13)

10000
01000
001200
00210
000012
,
10000
01000
00100
0011120
001012
,
10000
01000
0011110
00821
005120
,
29000
92000
001200
000120
000012
,
36000
710000
00100
00501
00810

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

Export

Character table of A4⋊Dic6 in TeX

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