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G = D12.29D6  order 288 = 25·32

4th non-split extension by D12 of D6 acting via D6/C6=C2

metabelian, supersoluble, monomial

Aliases: D12.29D6, C12.75D12, C62.50D4, Dic6.28D6, C3⋊C8.3D6, C4○D12.5S3, C6.70(C2×D12), (C3×C12).68D4, (C2×C6).14D12, C4.Dic3⋊9S3, (C2×C12).120D6, C3⋊5(C8.D6), C32⋊3Q16⋊5C2, D12.S3⋊4C2, C3⋊1(Q8.14D6), C12.48(C3⋊D4), (C6×C12).80C22, (C3×C12).67C23, C12.84(C22×S3), C32⋊7(C8.C22), C4.17(C3⋊D12), (C3×D12).37C22, (C3×Dic6).35C22, C22.10(C3⋊D12), C32⋊4Q8.27C22, (C2×C4).9S32, C4.55(C2×S32), C6.6(C2×C3⋊D4), (C3×C6).71(C2×D4), (C3×C3⋊C8).3C22, (C3×C4○D12).8C2, (C3×C4.Dic3)⋊3C2, C2.10(C2×C3⋊D12), (C2×C6).20(C3⋊D4), (C2×C32⋊4Q8)⋊10C2, SmallGroup(288,479)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C12 — D12.29D6
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — D12.S3 — D12.29D6
Lower central
C32 — C3×C6 — C3×C12 — D12.29D6
Upper central
C1 — C2 — C2×C4

Generators and relations for D12.29D6
 G = < a,b,c,d | a12=b2=1, c6=a6, d2=a9, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=a6c5 >

Subgroups: 514 in 139 conjugacy classes, 44 normal (34 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8, C24, Dic6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C8.C22, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C62, C24⋊C2, Dic12, C4.Dic3, D4.S3, C3⋊Q16, C3×M4(2), C2×Dic6, C4○D12, C3×C4○D4, C3×C3⋊C8, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C32⋊4Q8, C32⋊4Q8, C2×C3⋊Dic3, C6×C12, C8.D6, Q8.14D6, D12.S3, C32⋊3Q16, C3×C4.Dic3, C3×C4○D12, C2×C32⋊4Q8, D12.29D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C3⋊D4, C22×S3, C8.C22, S32, C2×D12, C2×C3⋊D4, C3⋊D12, C2×S32, C8.D6, Q8.14D6, C2×C3⋊D12, D12.29D6

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

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

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

39 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D4E6A6B6C···6G6H6I8A8B12A12B12C12D12E···12J12K12L24A24B24C24D
order122233344444666···666881212121212···12121224242424
size1121222422123636224···41212121222224···4121212121212

39 irreducible representations

dim11111122222222222244444444
type++++++++++++++++-++++---
imageC1C2C2C2C2C2S3S3D4D4D6D6D6D6D12C3⋊D4D12C3⋊D4C8.C22S32C3⋊D12C2×S32C3⋊D12C8.D6Q8.14D6D12.29D6
kernelD12.29D6D12.S3C32⋊3Q16C3×C4.Dic3C3×C4○D12C2×C32⋊4Q8C4.Dic3C4○D12C3×C12C62C3⋊C8Dic6D12C2×C12C12C12C2×C6C2×C6C32C2×C4C4C4C22C3C3C1
# reps12211111112112222211111224

Matrix representation of D12.29D6 ►in GL4(𝔽73) generated by

146600
7700
70766
066714
,
22223941
1010412
60705163
70485163
,
59700
666600
075966
660766
,
77520
14145252
56555966
56565966
G:=sub<GL(4,GF(73))| [14,7,7,0,66,7,0,66,0,0,7,7,0,0,66,14],[22,10,60,70,22,10,70,48,39,41,51,51,41,2,63,63],[59,66,0,66,7,66,7,0,0,0,59,7,0,0,66,66],[7,14,56,56,7,14,55,56,52,52,59,59,0,52,66,66] >;
 

D12.29D6 in GAP, Magma, Sage, TeX 

D_{12}._{29}D_6
 
% in TeX
 
G:=Group("D12.29D6");
 
// GroupNames label
 
G:=SmallGroup(288,479);
 
// by ID
 
G=gap.SmallGroup(288,479);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,64,219,100,675,80,1356,9414]);
 
// 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,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=a^6*c^5>;
 
// generators/relations
 

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