Copied to
clipboard

G = D12.10D6  order 288 = 25·32

10th non-split extension by D12 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: D12.10D6, Q8:5S32, C3:C8:11D6, (C3xQ8):6D6, C6.62(S3xD4), D6:D6:5C2, C3:4(Q8:3D6), Q8:2S3:3S3, C3:D24:14C2, C3:Dic3.59D4, C12:S3:6C22, C12.31D6:4C2, C12.26D6:1C2, C32:15(C8:C22), C12.18(C22xS3), (C3xC12).18C23, C2.22(Dic3:D6), (Q8xC32):4C22, (C3xD12).18C22, C4.18(C2xS32), (C3xC3:C8):9C22, (C2xC3:S3).24D4, (C3xQ8:2S3):2C2, (C3xC6).133(C2xD4), (C4xC3:S3).18C22, SmallGroup(288,589)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3xC12 — D12.10D6
Chief series
C1 — C3 — C32 — C3xC6 — C3xC12 — C3xD12 — D6:D6 — D12.10D6
Lower central
C32 — C3xC6 — C3xC12 — D12.10D6
Upper central
C1 — C2 — C4 — Q8

Generators and relations for D12.10D6
 G = < a,b,c,d | a12=b2=1, c6=a6, d2=a3, bab=a-1, cac-1=a7, ad=da, cbc-1=dbd-1=a3b, dcd-1=a9c5 >

Subgroups: 826 in 156 conjugacy classes, 38 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2xC4, D4, Q8, C23, C32, Dic3, C12, C12, D6, C2xC6, M4(2), D8, SD16, C2xD4, C4oD4, C3xS3, C3:S3, C3xC6, C3:C8, C24, C4xS3, D12, D12, C3:D4, C3xD4, C3xQ8, C3xQ8, C22xS3, C8:C22, C3:Dic3, C3xC12, C3xC12, S32, S3xC6, C2xC3:S3, C2xC3:S3, C8:S3, D24, D4:S3, Q8:2S3, C3xSD16, S3xD4, Q8:3S3, C3xC3:C8, D6:S3, C3xD12, C4xC3:S3, C4xC3:S3, C12:S3, C12:S3, Q8xC32, C2xS32, Q8:3D6, C12.31D6, C3:D24, C3xQ8:2S3, D6:D6, C12.26D6, D12.10D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C22xS3, C8:C22, S32, S3xD4, C2xS32, Q8:3D6, Dic3:D6, D12.10D6

Character table of D12.10D6

 class 12A2B2C2D2E3A3B3C4A4B4C6A6B6C6D6E8A8B12A12B12C12D12E12F12G24A24B24C24D
 size 1112121836224241822424241212448888812121212
ρ1111111111111111111111111111111    trivial
ρ211-11-111111-1-1111-11-1111-1-11-1-1-111-1    linear of order 2
ρ311111-11111-1111111-1-111-1-11-1-1-1-1-1-1    linear of order 2
ρ411-11-1-111111-1111-111-111111111-1-11    linear of order 2
ρ511-1-11-11111-11111-1-11111-1-11-1-11111    linear of order 2
ρ6111-1-1-111111-11111-1-111111111-111-1    linear of order 2
ρ711-1-111111111111-1-1-1-11111111-1-1-1-1    linear of order 2
ρ8111-1-111111-1-11111-11-111-1-11-1-11-1-11    linear of order 2
ρ9222000-12-12202-1-1-1002-122-1-1-1-10-1-10    orthogonal lifted from S3
ρ1022-2000-12-12-202-1-11002-12-21-1110-1-10    orthogonal lifted from D6
ρ11220-2002-1-12-20-12-101202-111-11-2-100-1    orthogonal lifted from D6
ρ1222-2000-12-12202-1-1100-2-122-1-1-1-10110    orthogonal lifted from D6
ρ13222000-12-12-202-1-1-100-2-12-21-1110110    orthogonal lifted from D6
ρ142202002-1-1220-12-10-1202-1-1-1-1-12-100-1    orthogonal lifted from S3
ρ15220-2002-1-1220-12-101-202-1-1-1-1-121001    orthogonal lifted from D6
ρ162202002-1-12-20-12-10-1-202-111-11-21001    orthogonal lifted from D6
ρ17220020222-20-22220000-2-200-2000000    orthogonal lifted from D4
ρ182200-20222-2022220000-2-200-2000000    orthogonal lifted from D4
ρ19440000-2-214-40-2-210000-2-22-11-120000    orthogonal lifted from C2xS32
ρ20440000-24-2-4004-2-200002-4002000000    orthogonal lifted from S3xD4
ρ21440000-2-21-400-2-210000220-3-1300000    orthogonal lifted from Dic3:D6
ρ224400004-2-2-400-24-20000-42002000000    orthogonal lifted from S3xD4
ρ23440000-2-21440-2-210000-2-2-2111-20000    orthogonal lifted from S32
ρ244-40000444000-4-4-4000000000000000    orthogonal lifted from C8:C22
ρ25440000-2-21-400-2-2100002203-1-300000    orthogonal lifted from Dic3:D6
ρ264-40000-24-2000-422000000000000√6-√60    orthogonal lifted from Q8:3D6
ρ274-40000-24-2000-422000000000000-√6√60    orthogonal lifted from Q8:3D6
ρ284-400004-2-20002-4200000000000√600-√6    orthogonal lifted from Q8:3D6
ρ294-400004-2-20002-4200000000000-√600√6    orthogonal lifted from Q8:3D6
ρ308-80000-4-4200044-2000000000000000    orthogonal faithful, Schur index 2

Smallest permutation representation of D12.10D6
►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 39)(14 38)(15 37)(16 48)(17 47)(18 46)(19 45)(20 44)(21 43)(22 42)(23 41)(24 40)

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

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

Matrix representation of D12.10D6 ►in GL8(F73)

072000000
172000000
000720000
001720000
0000721271
000072020
0000721172
000072010
,
001720000
000720000
172000000
072000000
000090240
00009642449
0000210640
00002152649
,
172000000
10000000
00010000
007210000
000041644556
00009321728
00002719329
000054466441
,
00010000
007210000
172000000
10000000
00006810563
00006351068
000034500
0000683900

G:=sub<GL(8,GF(73))| [0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,72,72,72,72,0,0,0,0,1,0,1,0,0,0,0,0,2,2,1,1,0,0,0,0,71,0,72,0],[0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,9,9,21,21,0,0,0,0,0,64,0,52,0,0,0,0,24,24,64,64,0,0,0,0,0,49,0,9],[1,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,41,9,27,54,0,0,0,0,64,32,19,46,0,0,0,0,45,17,32,64,0,0,0,0,56,28,9,41],[0,0,1,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,68,63,34,68,0,0,0,0,10,5,5,39,0,0,0,0,5,10,0,0,0,0,0,0,63,68,0,0] >;
 

D12.10D6 in GAP, Magma, Sage, TeX 

D_{12}._{10}D_6
 
% in TeX
 
G:=Group("D12.10D6");
 
// GroupNames label
 
G:=SmallGroup(288,589);
 
// by ID
 
G=gap.SmallGroup(288,589);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,303,100,675,346,185,80,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^12=b^2=1,c^6=a^6,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^3*b,d*c*d^-1=a^9*c^5>;
 
// generators/relations
 

Export

Character table of D12.10D6 in TeX

׿
x
:
Z
F
o
wr
Q
<