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G = D12⋊9D6  order 288 = 25·32

3rd semidirect product of D12 and D6 acting via D6/S3=C2

metabelian, supersoluble, monomial

Aliases: D12⋊9D6, C3⋊C8⋊7D6, D4.3S32, D4⋊S3⋊4S3, (S3×D4)⋊2S3, (C4×S3).7D6, C3⋊6(D8⋊S3), (C3×D4).11D6, (S3×C6).33D4, C6.150(S3×D4), C32⋊2D8⋊7C2, D12⋊5S3⋊4C2, (C3×D12)⋊7C22, D6.Dic3⋊6C2, C12.9(C22×S3), (C3×C12).9C23, D12.S3⋊8C2, C32⋊9SD16⋊3C2, C3⋊2(D12⋊6C22), D6.13(C3⋊D4), C32⋊11(C8⋊C22), C32⋊4C8⋊6C22, (C3×Dic3).13D4, C32⋊4Q8⋊5C22, (S3×C12).14C22, (D4×C32).5C22, Dic3.10(C3⋊D4), C4.9(C2×S32), (C3×S3×D4)⋊2C2, (C3×D4⋊S3)⋊7C2, (C3×C3⋊C8)⋊12C22, C2.24(S3×C3⋊D4), C6.46(C2×C3⋊D4), (C3×C6).124(C2×D4), SmallGroup(288,580)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C12 — D12⋊9D6
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — D12⋊5S3 — D12⋊9D6
Lower central
C32 — C3×C6 — C3×C12 — D12⋊9D6
Upper central
C1 — C2 — C4 — D4

Generators and relations for D12⋊9D6
 G = < a,b,c,d | a12=b2=c6=d2=1, bab=a-1, cac-1=dad=a7, cbc-1=a3b, dbd=a9b, dcd=c-1 >

Subgroups: 586 in 146 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, D4, D4, Q8, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, M4(2), D8, SD16, C2×D4, C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8, C3⋊C8, C24, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×D4, C22×S3, C22×C6, C8⋊C22, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C62, C8⋊S3, C24⋊C2, C4.Dic3, D4⋊S3, D4⋊S3, D4.S3, C3×D8, C4○D12, S3×D4, D4⋊2S3, C6×D4, C3×C3⋊C8, C32⋊4C8, S3×Dic3, D6⋊S3, S3×C12, C3×D12, C3×C3⋊D4, C32⋊4Q8, D4×C32, S3×C2×C6, D8⋊S3, D12⋊6C22, D6.Dic3, C32⋊2D8, D12.S3, C3×D4⋊S3, C32⋊9SD16, D12⋊5S3, C3×S3×D4, D12⋊9D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C8⋊C22, S32, S3×D4, C2×C3⋊D4, C2×S32, D8⋊S3, D12⋊6C22, S3×C3⋊D4, 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 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(25 42)(26 41)(27 40)(28 39)(29 38)(30 37)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)

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

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

33 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C6A6B6C6D6E6F6G6H6I6J6K6L6M8A8B12A12B12C12D24A24B
order122222333444666666666666688121212122424
size11461212224263622444668881212241236448121212

33 irreducible representations

dim11111111222222222244444448
type++++++++++++++++++++-
imageC1C2C2C2C2C2C2C2S3S3D4D4D6D6D6D6C3⋊D4C3⋊D4C8⋊C22S32S3×D4C2×S32D8⋊S3D12⋊6C22S3×C3⋊D4D12⋊9D6
kernelD12⋊9D6D6.Dic3C32⋊2D8D12.S3C3×D4⋊S3C32⋊9SD16D12⋊5S3C3×S3×D4D4⋊S3S3×D4C3×Dic3S3×C6C3⋊C8C4×S3D12C3×D4Dic3D6C32D4C6C4C3C3C2C1
# reps11111111111111222211112221

Matrix representation of D12⋊9D6 ►in GL8(𝔽73)

10000000
01000000
006500000
00090000
00000100
000072000
0000727212
0000017272
,
720000000
072000000
00010000
00100000
0000727212
00000010
00000100
000007211
,
172000000
10000000
00100000
000720000
00001000
000007200
0000727212
000010072
,
10000000
172000000
00100000
000720000
000037426262
00003136011
00003737510
00000363168

G:=sub<GL(8,GF(73))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,65,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,72,1,0,0,0,0,0,0,1,72,0,0,0,0,0,0,2,72],[72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,1,72,0,0,0,0,1,1,0,1,0,0,0,0,2,0,0,1],[1,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,72,1,0,0,0,0,0,72,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,72],[1,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,37,31,37,0,0,0,0,0,42,36,37,36,0,0,0,0,62,0,5,31,0,0,0,0,62,11,10,68] >;
 

D12⋊9D6 in GAP, Magma, Sage, TeX 

D_{12}\rtimes_9D_6
 
% in TeX
 
G:=Group("D12:9D6");
 
// GroupNames label
 
G:=SmallGroup(288,580);
 
// by ID
 
G=gap.SmallGroup(288,580);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,135,346,185,80,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^12=b^2=c^6=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^7,c*b*c^-1=a^3*b,d*b*d=a^9*b,d*c*d=c^-1>;
 
// generators/relations
 

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