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G = D12.28D6order 288 = 25·32

3rd non-split extension by D12 of D6 acting via D6/C6=C2

metabelian, supersoluble, monomial

Aliases: D12.28D6, C12.23D12, C62.49D4, C3⋊C84D6, (C2×D12)⋊9S3, (C6×D12)⋊4C2, C3⋊D246C2, C34(C8⋊D6), (C3×C12).67D4, C6.69(C2×D12), (C2×C6).60D12, C4.Dic38S3, (C2×C12).119D6, C328(C8⋊C22), D12.S33C2, C12.59D63C2, C31(D126C22), C12.47(C3⋊D4), (C6×C12).79C22, (C3×C12).66C23, C12⋊S317C22, C4.23(C3⋊D12), C12.126(C22×S3), (C3×D12).36C22, C324Q816C22, C22.4(C3⋊D12), (C2×C4).8S32, C4.54(C2×S32), (C3×C3⋊C8)⋊4C22, C6.5(C2×C3⋊D4), (C3×C6).70(C2×D4), C2.9(C2×C3⋊D12), (C3×C4.Dic3)⋊2C2, (C2×C6).19(C3⋊D4), SmallGroup(288,478)

Series: Derived Chief Lower central Upper central

C1C3×C12 — D12.28D6
C1C3C32C3×C6C3×C12C3×D12D12.S3 — D12.28D6
C32C3×C6C3×C12 — D12.28D6
C1C2C2×C4

Generators and relations for D12.28D6
 G = < a,b,c,d | a12=b2=c6=1, d2=a3, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=a6c-1 >

Subgroups: 698 in 156 conjugacy classes, 44 normal (32 characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4 [×2], C4, C22, C22 [×5], S3 [×6], C6 [×2], C6 [×6], C8 [×2], C2×C4, C2×C4, D4 [×5], Q8, C23, C32, Dic3 [×4], C12 [×4], C12 [×2], D6 [×8], C2×C6 [×2], C2×C6 [×5], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C3×S3 [×2], C3⋊S3, C3×C6, C3×C6, C3⋊C8 [×2], C24 [×2], Dic6 [×3], C4×S3 [×4], D12 [×2], D12 [×4], C3⋊D4 [×4], C2×C12 [×2], C2×C12, C3×D4 [×3], C22×S3, C22×C6, C8⋊C22, C3⋊Dic3, C3×C12 [×2], S3×C6 [×4], C2×C3⋊S3, C62, C24⋊C2 [×2], D24 [×2], C4.Dic3, D4⋊S3 [×2], D4.S3 [×2], C3×M4(2), C2×D12, C4○D12 [×3], C6×D4, C3×C3⋊C8 [×2], C3×D12 [×2], C3×D12, C324Q8, C4×C3⋊S3, C12⋊S3, C327D4, C6×C12, S3×C2×C6, C8⋊D6, D126C22, C3⋊D24 [×2], D12.S3 [×2], C3×C4.Dic3, C6×D12, C12.59D6, D12.28D6
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, D126C22, C2×C3⋊D12, D12.28D6

Smallest permutation representation of D12.28D6
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 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 48)(8 47)(9 46)(10 45)(11 44)(12 43)(13 36)(14 35)(15 34)(16 33)(17 32)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)
(1 15 5 19 9 23)(2 16 6 20 10 24)(3 17 7 21 11 13)(4 18 8 22 12 14)(25 41 33 37 29 45)(26 42 34 38 30 46)(27 43 35 39 31 47)(28 44 36 40 32 48)
(1 42 4 45 7 48 10 39)(2 43 5 46 8 37 11 40)(3 44 6 47 9 38 12 41)(13 30 16 33 19 36 22 27)(14 31 17 34 20 25 23 28)(15 32 18 35 21 26 24 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), (1,42)(2,41)(3,40)(4,39)(5,38)(6,37)(7,48)(8,47)(9,46)(10,45)(11,44)(12,43)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25), (1,15,5,19,9,23)(2,16,6,20,10,24)(3,17,7,21,11,13)(4,18,8,22,12,14)(25,41,33,37,29,45)(26,42,34,38,30,46)(27,43,35,39,31,47)(28,44,36,40,32,48), (1,42,4,45,7,48,10,39)(2,43,5,46,8,37,11,40)(3,44,6,47,9,38,12,41)(13,30,16,33,19,36,22,27)(14,31,17,34,20,25,23,28)(15,32,18,35,21,26,24,29)>;

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

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

39 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C6A6B6C6D6E6F6G6H6I6J6K6L8A8B12A12B12C···12I24A24B24C24D
order12222233344466666666666688121212···1224242424
size112121236224223622224444121212121212224···412121212

39 irreducible representations

dim1111112222222222244444444
type+++++++++++++++++++++
imageC1C2C2C2C2C2S3S3D4D4D6D6D6D12C3⋊D4D12C3⋊D4C8⋊C22S32C3⋊D12C2×S32C3⋊D12C8⋊D6D126C22D12.28D6
kernelD12.28D6C3⋊D24D12.S3C3×C4.Dic3C6×D12C12.59D6C4.Dic3C2×D12C3×C12C62C3⋊C8D12C2×C12C12C12C2×C6C2×C6C32C2×C4C4C4C22C3C3C1
# reps1221111111222222211111224

Matrix representation of D12.28D6 in GL4(𝔽73) generated by

76600
71400
00766
00714
,
002727
00046
464600
02700
,
603000
433000
003043
003060
,
004330
004313
02700
464600
G:=sub<GL(4,GF(73))| [7,7,0,0,66,14,0,0,0,0,7,7,0,0,66,14],[0,0,46,0,0,0,46,27,27,0,0,0,27,46,0,0],[60,43,0,0,30,30,0,0,0,0,30,30,0,0,43,60],[0,0,0,46,0,0,27,46,43,43,0,0,30,13,0,0] >;

D12.28D6 in GAP, Magma, Sage, TeX

D_{12}._{28}D_6
% in TeX

G:=Group("D12.28D6");
// GroupNames label

G:=SmallGroup(288,478);
// by ID

G=gap.SmallGroup(288,478);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,64,219,675,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^6=1,d^2=a^3,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^-1>;
// generators/relations

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