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G = D126D6order 288 = 25·32

6th semidirect product of D12 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: D126D6, Q83S32, C3⋊C810D6, (C3×Q8)⋊4D6, (S3×D12)⋊6C2, (C4×S3).9D6, C37(Q83D6), Q83S34S3, Q82S31S3, (S3×C6).12D4, C6.155(S3×D4), C322D88C2, C33(D4⋊D6), C3⋊D2413C2, D6.8(C3⋊D4), (C3×D12)⋊9C22, D6.Dic33C2, C3211SD162C2, C3214(C8⋊C22), (C3×C12).16C23, C12.16(C22×S3), C324C89C22, (C3×Dic3).36D4, (Q8×C32)⋊2C22, (S3×C12).19C22, C12⋊S3.11C22, Dic3.17(C3⋊D4), C4.16(C2×S32), (C3×C3⋊C8)⋊8C22, C2.29(S3×C3⋊D4), C6.51(C2×C3⋊D4), (C3×Q83S3)⋊1C2, (C3×Q82S3)⋊1C2, (C3×C6).131(C2×D4), SmallGroup(288,587)

Series: Derived Chief Lower central Upper central

C1C3×C12 — D126D6
C1C3C32C3×C6C3×C12S3×C12S3×D12 — D126D6
C32C3×C6C3×C12 — D126D6
C1C2C4Q8

Generators and relations for D126D6
 G = < a,b,c,d | a12=b2=c6=d2=1, bab=dad=a-1, cac-1=a5, cbc-1=a10b, dbd=a7b, dcd=c-1 >

Subgroups: 714 in 146 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4, C4 [×2], C22 [×6], S3 [×6], C6 [×2], C6 [×4], C8 [×2], C2×C4 [×2], D4 [×5], Q8, C23, C32, Dic3, C12 [×2], C12 [×5], D6, D6 [×9], C2×C6 [×3], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C3×S3 [×3], C3⋊S3, C3×C6, C3⋊C8, C3⋊C8 [×3], C24, C4×S3, C4×S3, D12 [×2], D12 [×5], C3⋊D4, C2×C12 [×2], C3×D4 [×3], C3×Q8 [×2], C3×Q8, C22×S3 [×2], C8⋊C22, C3×Dic3, C3×C12, C3×C12, S32 [×2], S3×C6, S3×C6 [×2], C2×C3⋊S3, C8⋊S3, D24, C4.Dic3, D4⋊S3 [×3], Q82S3, Q82S3 [×3], C3×SD16, C2×D12, S3×D4, Q83S3, C3×C4○D4, C3×C3⋊C8, C324C8, C3⋊D12, S3×C12, S3×C12, C3×D12 [×2], C3×D12, C12⋊S3, Q8×C32, C2×S32, Q83D6, D4⋊D6, D6.Dic3, C322D8, C3⋊D24, C3×Q82S3, C3211SD16, S3×D12, C3×Q83S3, D126D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C3⋊D4 [×2], C22×S3 [×2], C8⋊C22, S32, S3×D4, C2×C3⋊D4, C2×S32, Q83D6, D4⋊D6, S3×C3⋊D4, D126D6

Smallest permutation representation of D126D6
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 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 40)(10 39)(11 38)(12 37)(13 27)(14 26)(15 25)(16 36)(17 35)(18 34)(19 33)(20 32)(21 31)(22 30)(23 29)(24 28)
(1 35 5 31 9 27)(2 28 6 36 10 32)(3 33 7 29 11 25)(4 26 8 34 12 30)(13 38 17 46 21 42)(14 43 18 39 22 47)(15 48 19 44 23 40)(16 41 20 37 24 45)
(1 5)(2 4)(6 12)(7 11)(8 10)(13 14)(15 24)(16 23)(17 22)(18 21)(19 20)(25 33)(26 32)(27 31)(28 30)(34 36)(37 48)(38 47)(39 46)(40 45)(41 44)(42 43)

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

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

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

33 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C6A6B6C6D6E6F6G8A8B12A12B12C12D12E12F12G12H12I12J24A24B
order122222333444666666688121212121212121212122424
size11612123622424622412121224123644446688881212

33 irreducible representations

dim11111111222222222244444448
type+++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3S3D4D4D6D6D6D6C3⋊D4C3⋊D4C8⋊C22S32S3×D4C2×S32Q83D6D4⋊D6S3×C3⋊D4D126D6
kernelD126D6D6.Dic3C322D8C3⋊D24C3×Q82S3C3211SD16S3×D12C3×Q83S3Q82S3Q83S3C3×Dic3S3×C6C3⋊C8C4×S3D12C3×Q8Dic3D6C32Q8C6C4C3C3C2C1
# reps11111111111111222211112221

Matrix representation of D126D6 in GL8(ℤ)

000-10000
001-10000
01000000
-11000000
00000001
000000-11
00000-100
00001-100
,
00001001
00001-110
00000-110
0000-101-1
01-100000
10-110000
10010000
1-1100000
,
00001-100
00000-100
000000-11
00000001
-10000000
-11000000
00100000
001-10000
,
-10000000
-11000000
00100000
001-10000
00001-100
00000-100
000000-11
00000001

G:=sub<GL(8,Integers())| [0,0,0,-1,0,0,0,0,0,0,1,1,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,1,1,0,0],[0,0,0,0,0,1,1,1,0,0,0,0,1,0,0,-1,0,0,0,0,-1,-1,0,1,0,0,0,0,0,1,1,0,1,1,0,-1,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,1,1,0,0,0,0,1,0,0,-1,0,0,0,0],[0,0,0,0,-1,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,1,0,0,0,0],[-1,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,1] >;

D126D6 in GAP, Magma, Sage, TeX

D_{12}\rtimes_6D_6
% in TeX

G:=Group("D12:6D6");
// GroupNames label

G:=SmallGroup(288,587);
// by ID

G=gap.SmallGroup(288,587);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,422,135,100,346,185,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^6=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^10*b,d*b*d=a^7*b,d*c*d=c^-1>;
// generators/relations

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