Copied to
clipboard

G = D12⋊6D6  order 288 = 25·32

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

metabelian, supersoluble, monomial

Aliases: D12⋊6D6, Q8⋊3S32, C3⋊C8⋊10D6, (C3×Q8)⋊4D6, (S3×D12)⋊6C2, (C4×S3).9D6, C3⋊7(Q8⋊3D6), Q8⋊3S3⋊4S3, Q8⋊2S3⋊1S3, (S3×C6).12D4, C6.155(S3×D4), C32⋊2D8⋊8C2, C3⋊3(D4⋊D6), C3⋊D24⋊13C2, D6.8(C3⋊D4), (C3×D12)⋊9C22, D6.Dic3⋊3C2, C32⋊11SD16⋊2C2, C32⋊14(C8⋊C22), (C3×C12).16C23, C12.16(C22×S3), C32⋊4C8⋊9C22, (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×Q8⋊3S3)⋊1C2, (C3×Q8⋊2S3)⋊1C2, (C3×C6).131(C2×D4), SmallGroup(288,587)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C12 — D12⋊6D6
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — S3×D12 — D12⋊6D6
Lower central
C32 — C3×C6 — C3×C12 — D12⋊6D6
Upper central
C1 — C2 — C4 — Q8

Generators and relations for D12⋊6D6
 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, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, D4, Q8, C23, C32, Dic3, C12, C12, D6, D6, C2×C6, M4(2), D8, SD16, C2×D4, C4○D4, C3×S3, C3⋊S3, C3×C6, C3⋊C8, C3⋊C8, C24, C4×S3, C4×S3, D12, D12, C3⋊D4, C2×C12, C3×D4, C3×Q8, C3×Q8, C22×S3, C8⋊C22, C3×Dic3, C3×C12, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, C8⋊S3, D24, C4.Dic3, D4⋊S3, Q8⋊2S3, Q8⋊2S3, C3×SD16, C2×D12, S3×D4, Q8⋊3S3, C3×C4○D4, C3×C3⋊C8, C32⋊4C8, C3⋊D12, S3×C12, S3×C12, C3×D12, C3×D12, C12⋊S3, Q8×C32, C2×S32, Q8⋊3D6, D4⋊D6, D6.Dic3, C32⋊2D8, C3⋊D24, C3×Q8⋊2S3, C32⋊11SD16, S3×D12, C3×Q8⋊3S3, D12⋊6D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C8⋊C22, S32, S3×D4, C2×C3⋊D4, C2×S32, Q8⋊3D6, D4⋊D6, S3×C3⋊D4, D12⋊6D6

Smallest permutation representation of D12⋊6D6
►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 39)(2 38)(3 37)(4 48)(5 47)(6 46)(7 45)(8 44)(9 43)(10 42)(11 41)(12 40)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)

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

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

33 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C6A6B6C6D6E6F6G8A8B12A12B12C12D12E12F12G12H12I12J24A24B
order122222333444666666688121212121212121212122424
size11612123622424622412121224123644446688881212

33 irreducible representations

dim11111111222222222244444448
type+++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3S3D4D4D6D6D6D6C3⋊D4C3⋊D4C8⋊C22S32S3×D4C2×S32Q8⋊3D6D4⋊D6S3×C3⋊D4D12⋊6D6
kernelD12⋊6D6D6.Dic3C32⋊2D8C3⋊D24C3×Q8⋊2S3C32⋊11SD16S3×D12C3×Q8⋊3S3Q8⋊2S3Q8⋊3S3C3×Dic3S3×C6C3⋊C8C4×S3D12C3×Q8Dic3D6C32Q8C6C4C3C3C2C1
# reps11111111111111222211112221

Matrix representation of D12⋊6D6 ►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] >;
 

D12⋊6D6 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
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁