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, (C3×Q8)⋊6D6, C6.62(S3×D4), 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(C22×S3), (C3×C12).18C23, C2.22(Dic3⋊D6), (Q8×C32)⋊4C22, (C3×D12).18C22, C4.18(C2×S32), (C3×C3⋊C8)⋊9C22, (C2×C3⋊S3).24D4, (C3×Q8⋊2S3)⋊2C2, (C3×C6).133(C2×D4), (C4×C3⋊S3).18C22, SmallGroup(288,589)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C12 — D12.10D6
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — D6⋊D6 — D12.10D6
Lower central
C32 — C3×C6 — C3×C12 — 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 [×4], C3 [×2], C3, C4, C4 [×2], C22 [×6], S3 [×9], C6 [×2], C6 [×3], C8 [×2], C2×C4 [×2], D4 [×5], Q8, C23, C32, Dic3 [×3], C12 [×2], C12 [×5], D6 [×13], C2×C6 [×2], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C3×S3 [×2], C3⋊S3 [×2], C3×C6, C3⋊C8 [×2], C24 [×2], C4×S3 [×7], D12 [×2], D12 [×7], C3⋊D4 [×2], C3×D4 [×2], C3×Q8 [×2], C3×Q8, C22×S3 [×2], C8⋊C22, C3⋊Dic3, C3×C12, C3×C12, S32 [×2], S3×C6 [×2], C2×C3⋊S3, C2×C3⋊S3, C8⋊S3 [×2], D24 [×2], D4⋊S3 [×2], Q8⋊2S3 [×2], C3×SD16 [×2], S3×D4 [×2], Q8⋊3S3 [×3], C3×C3⋊C8 [×2], D6⋊S3, C3×D12 [×2], C4×C3⋊S3, C4×C3⋊S3, C12⋊S3, C12⋊S3, Q8×C32, C2×S32, Q8⋊3D6 [×2], C12.31D6, C3⋊D24 [×2], C3×Q8⋊2S3 [×2], D6⋊D6, C12.26D6, D12.10D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C22×S3 [×2], C8⋊C22, S32, S3×D4 [×2], C2×S32, Q8⋊3D6 [×2], 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 C2×S32
ρ20440000-24-2-4004-2-200002-4002000000    orthogonal lifted from S3×D4
ρ21440000-2-21-400-2-210000220-3-1300000    orthogonal lifted from Dic3⋊D6
ρ224400004-2-2-400-24-20000-42002000000    orthogonal lifted from S3×D4
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 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 36)(11 35)(12 34)(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 21 11 19 9 17 7 15 5 13 3 23)(2 16 12 14 10 24 8 22 6 20 4 18)(25 44 27 46 29 48 31 38 33 40 35 42)(26 39 28 41 30 43 32 45 34 47 36 37)

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

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

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁