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G = D12.29D6order 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.Dic39S3, (C2×C12).120D6, C35(C8.D6), C323Q165C2, D12.S34C2, C31(Q8.14D6), C12.48(C3⋊D4), (C6×C12).80C22, (C3×C12).67C23, C12.84(C22×S3), C327(C8.C22), C4.17(C3⋊D12), (C3×D12).37C22, (C3×Dic6).35C22, C22.10(C3⋊D12), C324Q8.27C22, (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×C324Q8)⋊10C2, SmallGroup(288,479)

Series: Derived Chief Lower central Upper central

C1C3×C12 — D12.29D6
C1C3C32C3×C6C3×C12C3×D12D12.S3 — D12.29D6
C32C3×C6C3×C12 — D12.29D6
C1C2C2×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 [×2], C3 [×2], C3, C4 [×2], C4 [×3], C22, C22, S3, C6 [×2], C6 [×6], C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], Q8 [×4], C32, Dic3 [×9], C12 [×4], C12 [×3], D6, C2×C6 [×2], C2×C6 [×2], M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8 [×2], C24 [×2], Dic6, Dic6 [×10], C4×S3, D12, C2×Dic3 [×4], C3⋊D4, C2×C12 [×2], C2×C12 [×2], C3×D4 [×2], C3×Q8, C8.C22, C3×Dic3, C3⋊Dic3 [×2], C3×C12 [×2], S3×C6, C62, C24⋊C2 [×2], Dic12 [×2], C4.Dic3, D4.S3 [×2], C3⋊Q16 [×2], C3×M4(2), C2×Dic6 [×3], C4○D12, C3×C4○D4, C3×C3⋊C8 [×2], C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C324Q8 [×2], C324Q8, C2×C3⋊Dic3, C6×C12, C8.D6, Q8.14D6, D12.S3 [×2], C323Q16 [×2], C3×C4.Dic3, C3×C4○D12, C2×C324Q8, D12.29D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, D12 [×2], C3⋊D4 [×2], C22×S3 [×2], C8.C22, S32, C2×D12, C2×C3⋊D4, C3⋊D12 [×2], 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 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 36)(11 35)(12 34)(13 40)(14 39)(15 38)(16 37)(17 48)(18 47)(19 46)(20 45)(21 44)(22 43)(23 42)(24 41)
(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 23 10 20 7 17 4 14)(2 24 11 21 8 18 5 15)(3 13 12 22 9 19 6 16)(25 37 34 46 31 43 28 40)(26 38 35 47 32 44 29 41)(27 39 36 48 33 45 30 42)

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

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

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

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.S3C323Q16C3×C4.Dic3C3×C4○D12C2×C324Q8C4.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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