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G = D12.9D6  order 288 = 25·32

9th non-split extension by D12 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: D12.9D6, Q8⋊4S32, C3⋊C8⋊17D6, (C3×Q8)⋊5D6, C3⋊S3⋊5SD16, C3⋊4(S3×SD16), C6.61(S3×D4), Q8⋊2S3⋊2S3, D6⋊D6.4C2, C3⋊Dic3.22D4, C32⋊12(C2×SD16), C12.29D6⋊5C2, C12.17(C22×S3), (C3×C12).17C23, D12.S3⋊13C2, C2.21(Dic3⋊D6), (Q8×C32)⋊3C22, C32⋊4Q8⋊6C22, (C3×D12).17C22, C4.17(C2×S32), (Q8×C3⋊S3)⋊1C2, (C3×C3⋊C8)⋊17C22, (C2×C3⋊S3).59D4, (C3×Q8⋊2S3)⋊6C2, (C3×C6).132(C2×D4), (C4×C3⋊S3).17C22, SmallGroup(288,588)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C12 — D12.9D6
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — D6⋊D6 — D12.9D6
Lower central
C32 — C3×C6 — C3×C12 — D12.9D6
Upper central
C1 — C2 — C4 — Q8

Generators and relations for D12.9D6
 G = < a,b,c,d | a12=b2=1, c6=a6, d2=a9, bab=a-1, cac-1=a7, ad=da, cbc-1=dbd-1=a3b, dcd-1=a3c5 >

Subgroups: 738 in 155 conjugacy classes, 40 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, D4, Q8, Q8, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C8, SD16, C2×D4, C2×Q8, C3×S3, C3⋊S3, C3×C6, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C3×D4, C3×Q8, C3×Q8, C22×S3, C2×SD16, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, S32, S3×C6, C2×C3⋊S3, S3×C8, C24⋊C2, D4.S3, Q8⋊2S3, C3×SD16, S3×D4, S3×Q8, C3×C3⋊C8, D6⋊S3, C3×D12, C32⋊4Q8, C32⋊4Q8, C4×C3⋊S3, C4×C3⋊S3, Q8×C32, C2×S32, S3×SD16, C12.29D6, D12.S3, C3×Q8⋊2S3, D6⋊D6, Q8×C3⋊S3, D12.9D6
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C22×S3, C2×SD16, S32, S3×D4, C2×S32, S3×SD16, Dic3⋊D6, D12.9D6

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

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

33 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D6A6B6C6D6E8A8B8C8D12A12B12C···12G24A24B24C24D
order1222223334444666668888121212···1224242424
size1199121222424183622424246666448···812121212

33 irreducible representations

dim1111112222222444448
type++++++++++++++++-
imageC1C2C2C2C2C2S3D4D4D6D6D6SD16S32S3×D4C2×S32S3×SD16Dic3⋊D6D12.9D6
kernelD12.9D6C12.29D6D12.S3C3×Q8⋊2S3D6⋊D6Q8×C3⋊S3Q8⋊2S3C3⋊Dic3C2×C3⋊S3C3⋊C8D12C3×Q8C3⋊S3Q8C6C4C3C2C1
# reps1122112112224121421

Matrix representation of D12.9D6 ►in GL6(𝔽73)

7210000
7200000
0072300
0048100
000010
000001
,
7200000
7210000
00626800
00241100
000010
000001
,
100000
010000
0062100
00336700
0000721
0000720
,
100000
010000
00125500
004000
0000720
0000721

G:=sub<GL(6,GF(73))| [72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,48,0,0,0,0,3,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,72,0,0,0,0,0,1,0,0,0,0,0,0,62,24,0,0,0,0,68,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,33,0,0,0,0,21,67,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,4,0,0,0,0,55,0,0,0,0,0,0,0,72,72,0,0,0,0,0,1] >;
 

D12.9D6 in GAP, Magma, Sage, TeX 

D_{12}._9D_6
 
% in TeX
 
G:=Group("D12.9D6");
 
// GroupNames label
 
G:=SmallGroup(288,588);
 
// by ID
 
G=gap.SmallGroup(288,588);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,135,100,675,346,185,80,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^12=b^2=1,c^6=a^6,d^2=a^9,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^3*c^5>;
 
// generators/relations
 

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