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G = D12.9D6order 288 = 25·32

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

metabelian, supersoluble, monomial

Aliases: D12.9D6, Q84S32, C3⋊C817D6, (C3×Q8)⋊5D6, C3⋊S35SD16, C34(S3×SD16), C6.61(S3×D4), Q82S32S3, D6⋊D6.4C2, C3⋊Dic3.22D4, C3212(C2×SD16), C12.29D65C2, C12.17(C22×S3), (C3×C12).17C23, D12.S313C2, C2.21(Dic3⋊D6), (Q8×C32)⋊3C22, C324Q86C22, (C3×D12).17C22, C4.17(C2×S32), (Q8×C3⋊S3)⋊1C2, (C3×C3⋊C8)⋊17C22, (C2×C3⋊S3).59D4, (C3×Q82S3)⋊6C2, (C3×C6).132(C2×D4), (C4×C3⋊S3).17C22, SmallGroup(288,588)

Series: Derived Chief Lower central Upper central

C1C3×C12 — D12.9D6
C1C3C32C3×C6C3×C12C3×D12D6⋊D6 — D12.9D6
C32C3×C6C3×C12 — D12.9D6
C1C2C4Q8

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 [×4], C3 [×2], C3, C4, C4 [×3], C22 [×5], S3 [×8], C6 [×2], C6 [×3], C8 [×2], C2×C4 [×2], D4 [×3], Q8, Q8 [×2], C23, C32, Dic3 [×7], C12 [×2], C12 [×5], D6 [×9], C2×C6 [×2], C2×C8, SD16 [×4], C2×D4, C2×Q8, C3×S3 [×2], C3⋊S3 [×2], C3×C6, C3⋊C8 [×2], C24 [×2], Dic6 [×7], C4×S3 [×7], D12 [×2], C3⋊D4 [×2], C3×D4 [×2], C3×Q8 [×2], C3×Q8, C22×S3 [×2], C2×SD16, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, S32 [×2], S3×C6 [×2], C2×C3⋊S3, S3×C8 [×2], C24⋊C2 [×2], D4.S3 [×2], Q82S3 [×2], C3×SD16 [×2], S3×D4 [×2], S3×Q8 [×3], C3×C3⋊C8 [×2], D6⋊S3, C3×D12 [×2], C324Q8, C324Q8, C4×C3⋊S3, C4×C3⋊S3, Q8×C32, C2×S32, S3×SD16 [×2], C12.29D6, D12.S3 [×2], C3×Q82S3 [×2], D6⋊D6, Q8×C3⋊S3, D12.9D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], SD16 [×2], C2×D4, C22×S3 [×2], C2×SD16, S32, S3×D4 [×2], C2×S32, S3×SD16 [×2], 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 27)(2 26)(3 25)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(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 15 11 13 9 23 7 21 5 19 3 17)(2 22 12 20 10 18 8 16 6 14 4 24)(25 47 27 37 29 39 31 41 33 43 35 45)(26 42 28 44 30 46 32 48 34 38 36 40)
(1 48 10 45 7 42 4 39)(2 37 11 46 8 43 5 40)(3 38 12 47 9 44 6 41)(13 27 22 36 19 33 16 30)(14 28 23 25 20 34 17 31)(15 29 24 26 21 35 18 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), (1,27)(2,26)(3,25)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43), (1,15,11,13,9,23,7,21,5,19,3,17)(2,22,12,20,10,18,8,16,6,14,4,24)(25,47,27,37,29,39,31,41,33,43,35,45)(26,42,28,44,30,46,32,48,34,38,36,40), (1,48,10,45,7,42,4,39)(2,37,11,46,8,43,5,40)(3,38,12,47,9,44,6,41)(13,27,22,36,19,33,16,30)(14,28,23,25,20,34,17,31)(15,29,24,26,21,35,18,32)>;

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

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

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×Q82S3D6⋊D6Q8×C3⋊S3Q82S3C3⋊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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