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G = Dic6.D6order 288 = 25·32

5th non-split extension by Dic6 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: Dic6.5D6, D4.8S32, C3⋊C8.6D6, D4.S33S3, C6.58(S3×D4), (C3×D4).10D6, C33(D4.D6), C3⋊Dic3.57D4, C12.8(C22×S3), (C3×C12).8C23, Dic3.D64C2, C12.31D63C2, C323Q1612C2, C2.18(Dic3⋊D6), C12.D6.2C2, C3210(C8.C22), (D4×C32).4C22, C324Q8.7C22, (C3×Dic6).11C22, C4.8(C2×S32), (C3×D4.S3)⋊2C2, (C2×C3⋊S3).22D4, (C3×C6).123(C2×D4), (C3×C3⋊C8).11C22, (C4×C3⋊S3).14C22, SmallGroup(288,579)

Series: Derived Chief Lower central Upper central

C1C3×C12 — Dic6.D6
C1C3C32C3×C6C3×C12C3×Dic6Dic3.D6 — Dic6.D6
C32C3×C6C3×C12 — Dic6.D6
C1C2C4D4

Generators and relations for Dic6.D6
 G = < a,b,c,d | a12=c6=1, b2=a6, d2=a3, bab-1=a-1, cac-1=a7, ad=da, cbc-1=a9b, dbd-1=a3b, dcd-1=a9c-1 >

Subgroups: 570 in 140 conjugacy classes, 38 normal (16 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4, C4 [×4], C22 [×2], S3 [×3], C6 [×2], C6 [×5], C8 [×2], C2×C4 [×3], D4, D4, Q8 [×4], C32, Dic3 [×9], C12 [×2], C12 [×3], D6 [×3], C2×C6 [×4], M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C3⋊S3, C3×C6, C3×C6, C3⋊C8 [×2], C24 [×2], Dic6 [×2], Dic6 [×5], C4×S3 [×5], C2×Dic3 [×4], C3⋊D4 [×4], C3×D4 [×2], C3×D4, C3×Q8 [×2], C8.C22, C3×Dic3 [×2], C3⋊Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, C8⋊S3 [×2], Dic12 [×2], D4.S3 [×2], C3⋊Q16 [×2], C3×SD16 [×2], D42S3 [×3], S3×Q8 [×2], C3×C3⋊C8 [×2], C6.D6, C322Q8, C3×Dic6 [×2], C324Q8, C4×C3⋊S3, C2×C3⋊Dic3, C327D4, D4×C32, D4.D6 [×2], C12.31D6, C323Q16 [×2], C3×D4.S3 [×2], Dic3.D6, C12.D6, Dic6.D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C22×S3 [×2], C8.C22, S32, S3×D4 [×2], C2×S32, D4.D6 [×2], Dic3⋊D6, Dic6.D6

Character table of Dic6.D6

 class 12A2B2C3A3B3C4A4B4C4D4E6A6B6C6D6E6F6G8A8B12A12B12C12D12E24A24B24C24D
 size 1141822421212183622488881212448242412121212
ρ1111111111111111111111111111111    trivial
ρ211-1-111111-1-11111-1-1-1-11-11111-111-1-1    linear of order 2
ρ3111-11111-11-1-111111111-1111-1111-1-1    linear of order 2
ρ411-111111-1-11-1111-1-1-1-111111-1-11111    linear of order 2
ρ511-111111111-1111-1-1-1-1-1-111111-1-1-1-1    linear of order 2
ρ6111-111111-1-1-11111111-111111-1-1-111    linear of order 2
ρ711-1-11111-11-11111-1-1-1-1-11111-11-1-111    linear of order 2
ρ811111111-1-1111111111-1-1111-1-1-1-1-1-1    linear of order 2
ρ92202222-200-20222000000-2-2-2000000    orthogonal lifted from D4
ρ1022-202-1-1220002-1-111-210-22-1-1-100011    orthogonal lifted from D6
ρ112220-12-120-200-12-1-1-1-12-20-12-1011100    orthogonal lifted from D6
ρ1222-20-12-120200-12-1111-2-20-12-10-11100    orthogonal lifted from D6
ρ1322-202-1-12-20002-1-111-21022-1-11000-1-1    orthogonal lifted from D6
ρ1422-20-12-120-200-12-1111-220-12-101-1-100    orthogonal lifted from D6
ρ152220-12-120200-12-1-1-1-1220-12-10-1-1-100    orthogonal lifted from S3
ρ1622202-1-12-20002-1-1-1-12-10-22-1-1100011    orthogonal lifted from D6
ρ17220-2222-20020222000000-2-2-2000000    orthogonal lifted from D4
ρ1822202-1-1220002-1-1-1-12-1022-1-1-1000-1-1    orthogonal lifted from S3
ρ194400-2-21-40000-2-21-33000022-1000000    orthogonal lifted from Dic3⋊D6
ρ204400-2-21-40000-2-213-3000022-1000000    orthogonal lifted from Dic3⋊D6
ρ214440-2-2140000-2-2111-2-200-2-21000000    orthogonal lifted from S32
ρ2244-40-2-2140000-2-21-1-12200-2-21000000    orthogonal lifted from C2×S32
ρ234400-24-2-40000-24-20000002-42000000    orthogonal lifted from S3×D4
ρ2444004-2-2-400004-2-2000000-422000000    orthogonal lifted from S3×D4
ρ254-40044400000-4-4-4000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ264-4004-2-200000-42200000000000006-6    symplectic lifted from D4.D6, Schur index 2
ρ274-4004-2-200000-4220000000000000-66    symplectic lifted from D4.D6, Schur index 2
ρ284-400-24-2000002-4200000000000-6600    symplectic lifted from D4.D6, Schur index 2
ρ294-400-24-2000002-42000000000006-600    symplectic lifted from D4.D6, Schur index 2
ρ308-800-4-420000044-2000000000000000    symplectic faithful, Schur index 2

Smallest permutation representation of Dic6.D6
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 27 7 33)(2 26 8 32)(3 25 9 31)(4 36 10 30)(5 35 11 29)(6 34 12 28)(13 40 19 46)(14 39 20 45)(15 38 21 44)(16 37 22 43)(17 48 23 42)(18 47 24 41)
(1 23 9 19 5 15)(2 18 10 14 6 22)(3 13 11 21 7 17)(4 20 12 16 8 24)(25 43 29 47 33 39)(26 38 30 42 34 46)(27 45 31 37 35 41)(28 40 32 44 36 48)
(1 47 4 38 7 41 10 44)(2 48 5 39 8 42 11 45)(3 37 6 40 9 43 12 46)(13 28 16 31 19 34 22 25)(14 29 17 32 20 35 23 26)(15 30 18 33 21 36 24 27)

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

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

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

Matrix representation of Dic6.D6 in GL8(𝔽73)

172000000
10000000
6700720000
06110000
0000727222
000010710
0000727211
000010720
,
676771720000
007210000
3636600000
3637600000
0000670260
0000664747
00007060
000066666767
,
721000000
720000000
82110000
2697200000
000065574619
00001685427
00001530816
000043585765
,
66210000
66120000
242567670000
252467670000
00006863510
00001056368
0000346800
000053900

G:=sub<GL(8,GF(73))| [1,1,67,0,0,0,0,0,72,0,0,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,72,1,72,1,0,0,0,0,72,0,72,0,0,0,0,0,2,71,1,72,0,0,0,0,2,0,1,0],[67,0,36,36,0,0,0,0,67,0,36,37,0,0,0,0,71,72,6,6,0,0,0,0,72,1,0,0,0,0,0,0,0,0,0,0,67,6,7,66,0,0,0,0,0,6,0,66,0,0,0,0,26,47,6,67,0,0,0,0,0,47,0,67],[72,72,8,2,0,0,0,0,1,0,2,69,0,0,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,65,16,15,43,0,0,0,0,57,8,30,58,0,0,0,0,46,54,8,57,0,0,0,0,19,27,16,65],[6,6,24,25,0,0,0,0,6,6,25,24,0,0,0,0,2,1,67,67,0,0,0,0,1,2,67,67,0,0,0,0,0,0,0,0,68,10,34,5,0,0,0,0,63,5,68,39,0,0,0,0,5,63,0,0,0,0,0,0,10,68,0,0] >;

Dic6.D6 in GAP, Magma, Sage, TeX

{\rm Dic}_6.D_6
% in TeX

G:=Group("Dic6.D6");
// GroupNames label

G:=SmallGroup(288,579);
// by ID

G=gap.SmallGroup(288,579);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,120,254,303,675,346,185,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^12=c^6=1,b^2=a^6,d^2=a^3,b*a*b^-1=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^9*b,d*b*d^-1=a^3*b,d*c*d^-1=a^9*c^-1>;
// generators/relations

Export

Character table of Dic6.D6 in TeX

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