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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, (C3xD4):2D6, C6.56(S3xD4), D6:D6:4C2, C3:3(D8:S3), C3:Dic3.56D4, (C3xC12).4C23, C12.4(C22xS3), D12.S3:7C2, C12.31D6:5C2, C12.D6:1C2, C32:10(C8:C22), (D4xC32):4C22, C2.16(Dic3:D6), C32:4Q8:3C22, (C3xD12).11C22, C4.4(C2xS32), (C3xD4:S3):6C2, (C3xC3:C8):11C22, (C2xC3:S3).21D4, (C3xC6).119(C2xD4), (C4xC3:S3).12C22, SmallGroup(288,575)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3xC12 — D12.D6
Chief series
C1 — C3 — C32 — C3xC6 — C3xC12 — C3xD12 — D6:D6 — D12.D6
Lower central
C32 — C3xC6 — C3xC12 — 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, C2xC4, D4, D4, Q8, C23, C32, Dic3, C12, C12, D6, C2xC6, M4(2), D8, SD16, C2xD4, C4oD4, C3xS3, C3:S3, C3xC6, C3xC6, C3:C8, C24, Dic6, C4xS3, D12, C2xDic3, C3:D4, C3xD4, C3xD4, C22xS3, C8:C22, C3:Dic3, C3:Dic3, C3xC12, S32, S3xC6, C2xC3:S3, C62, C8:S3, C24:C2, D4:S3, D4.S3, C3xD8, S3xD4, D4:2S3, C3xC3:C8, D6:S3, C3xD12, C32:4Q8, C4xC3:S3, C2xC3:Dic3, C32:7D4, D4xC32, C2xS32, D8:S3, C12.31D6, D12.S3, C3xD4:S3, D6:D6, C12.D6, D12.D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C22xS3, C8:C22, S32, S3xD4, C2xS32, 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 S3xD4
ρ224400004-2-2-4004-2-200000000-4220000    orthogonal lifted from S3xD4
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 C2xS32
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

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