Copied to
clipboard

G = D12.D6  order 288 = 25·32

5th non-split extension by D12 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: D12.5D6, D4⋊4S32, C3⋊C8⋊6D6, D4⋊S3⋊3S3, (C3×D4)⋊2D6, C6.56(S3×D4), D6⋊D6⋊4C2, C3⋊3(D8⋊S3), C3⋊Dic3.56D4, (C3×C12).4C23, C12.4(C22×S3), D12.S3⋊7C2, C12.31D6⋊5C2, C12.D6⋊1C2, C32⋊10(C8⋊C22), (D4×C32)⋊4C22, C2.16(Dic3⋊D6), C32⋊4Q8⋊3C22, (C3×D12).11C22, C4.4(C2×S32), (C3×D4⋊S3)⋊6C2, (C3×C3⋊C8)⋊11C22, (C2×C3⋊S3).21D4, (C3×C6).119(C2×D4), (C4×C3⋊S3).12C22, SmallGroup(288,575)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C12 — D12.D6
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — D6⋊D6 — D12.D6
Lower central
C32 — C3×C6 — C3×C12 — D12.D6
Upper central
C1 — C2 — C4 — D4

Generators and relations for D12.D6
 G = < a,b,c,d | a12=b2=c6=1, d2=a3, bab=a-1, cac-1=a7, ad=da, cbc-1=dbd-1=a9b, dcd-1=a3c-1 >

Subgroups: 762 in 156 conjugacy classes, 38 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, D4, D4, Q8, C23, C32, Dic3, C12, C12, D6, C2×C6, M4(2), D8, SD16, C2×D4, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C3×D4, C3×D4, C22×S3, C8⋊C22, C3⋊Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, C2×C3⋊S3, C62, C8⋊S3, C24⋊C2, D4⋊S3, D4.S3, C3×D8, S3×D4, D4⋊2S3, C3×C3⋊C8, D6⋊S3, C3×D12, C32⋊4Q8, C4×C3⋊S3, C2×C3⋊Dic3, C32⋊7D4, D4×C32, C2×S32, D8⋊S3, C12.31D6, D12.S3, C3×D4⋊S3, D6⋊D6, C12.D6, D12.D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C22×S3, C8⋊C22, S32, S3×D4, C2×S32, D8⋊S3, Dic3⋊D6, D12.D6

Character table of D12.D6

 class 12A2B2C2D2E3A3B3C4A4B4C6A6B6C6D6E6F6G6H6I8A8B12A12B12C24A24B24C24D
 size 1141212182242183622488882424121244812121212
ρ1111111111111111111111111111111    trivial
ρ2111-11-11111-1-111111111-11-1111-111-1    linear of order 2
ρ311-1-11-11111-11111-1-1-1-11-1-111111-1-11    linear of order 2
ρ411-111111111-1111-1-1-1-111-1-1111-1-1-1-1    linear of order 2
ρ511-11-1-11111-11111-1-1-1-1-111-1111-111-1    linear of order 2
ρ611-1-1-1111111-1111-1-1-1-1-1-1111111111    linear of order 2
ρ7111-1-111111111111111-1-1-1-1111-1-1-1-1    linear of order 2
ρ81111-1-11111-1-11111111-11-111111-1-11    linear of order 2
ρ922-2020-12-1200-12-111-21-10-20-12-10110    orthogonal lifted from D6
ρ102220-20-12-1200-12-1-1-12-110-20-12-10110    orthogonal lifted from D6
ρ11220002222-2-2022200000000-2-2-20000    orthogonal lifted from D4
ρ1222-22002-1-12002-1-1111-20-10-22-1-11001    orthogonal lifted from D6
ρ13222-2002-1-12002-1-1-1-1-12010-22-1-11001    orthogonal lifted from D6
ρ14222020-12-1200-12-1-1-12-1-1020-12-10-1-10    orthogonal lifted from S3
ρ1522000-2222-22022200000000-2-2-20000    orthogonal lifted from D4
ρ162222002-1-12002-1-1-1-1-120-1022-1-1-100-1    orthogonal lifted from S3
ρ1722-20-20-12-1200-12-111-211020-12-10-1-10    orthogonal lifted from D6
ρ1822-2-2002-1-12002-1-1111-201022-1-1-100-1    orthogonal lifted from D6
ρ19440000-2-21-400-2-213-300000022-10000    orthogonal lifted from Dic3⋊D6
ρ20440000-2-21-400-2-21-3300000022-10000    orthogonal lifted from Dic3⋊D6
ρ21440000-24-2-400-24-2000000002-420000    orthogonal lifted from S3×D4
ρ224400004-2-2-4004-2-200000000-4220000    orthogonal lifted from S3×D4
ρ23444000-2-21400-2-2111-2-20000-2-210000    orthogonal lifted from S32
ρ244-40000444000-4-4-4000000000000000    orthogonal lifted from C8⋊C22
ρ2544-4000-2-21400-2-21-1-1220000-2-210000    orthogonal lifted from C2×S32
ρ264-40000-24-20002-42000000000000√-6-√-60    complex lifted from D8⋊S3
ρ274-40000-24-20002-42000000000000-√-6√-60    complex lifted from D8⋊S3
ρ284-400004-2-2000-42200000000000√-600-√-6    complex lifted from D8⋊S3
ρ294-400004-2-2000-42200000000000-√-600√-6    complex lifted from D8⋊S3
ρ308-80000-4-4200044-2000000000000000    symplectic faithful, Schur index 2

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

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

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

Matrix representation of D12.D6 ►in GL8(𝔽73)

172000000
10000000
02110000
7107200000
00000100
000072000
00001112
00000727272
,
717171720000
717172710000
23220000
32220000
0000324600
0000464100
00004141689
0000032465
,
01000000
721000000
484672720000
5048100000
00000010
000072727271
00001000
00000001
,
717172710000
717171720000
5152220000
5251220000
000027275954
00003232564
0000464100
00000464114

G:=sub<GL(8,GF(73))| [1,1,0,71,0,0,0,0,72,0,2,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,1,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,2,72],[71,71,2,3,0,0,0,0,71,71,3,2,0,0,0,0,71,72,2,2,0,0,0,0,72,71,2,2,0,0,0,0,0,0,0,0,32,46,41,0,0,0,0,0,46,41,41,32,0,0,0,0,0,0,68,46,0,0,0,0,0,0,9,5],[0,72,48,50,0,0,0,0,1,1,46,48,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,71,0,1],[71,71,51,52,0,0,0,0,71,71,52,51,0,0,0,0,72,71,2,2,0,0,0,0,71,72,2,2,0,0,0,0,0,0,0,0,27,32,46,0,0,0,0,0,27,32,41,46,0,0,0,0,59,5,0,41,0,0,0,0,54,64,0,14] >;
 

D12.D6 in GAP, Magma, Sage, TeX 

D_{12}.D_6
 
% in TeX
 
G:=Group("D12.D6");
 
// GroupNames label
 
G:=SmallGroup(288,575);
 
// by ID
 
G=gap.SmallGroup(288,575);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,303,675,346,185,80,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^12=b^2=c^6=1,d^2=a^3,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=d*b*d^-1=a^9*b,d*c*d^-1=a^3*c^-1>;
 
// generators/relations
 

Export

Character table of D12.D6 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁