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G = D1212D6order 288 = 25·32

6th semidirect product of D12 and D6 acting via D6/S3=C2

metabelian, supersoluble, monomial

Aliases: D1212D6, C62.4C23, C3232+ 1+4, D47S32, (C4×S3)⋊3D6, (S3×D4)⋊4S3, C3⋊D44D6, (C3×D4)⋊10D6, (C22×S3)⋊6D6, D6⋊D612C2, C32(D46D6), (S3×C12)⋊7C22, D6.4D66C2, D6.D66C2, C6.20(S3×C23), (C3×C6).20C24, D125S310C2, C12.D68C2, (C3×D12)⋊14C22, (S3×C6).11C23, (C3×C12).32C23, C12.32(C22×S3), (S3×Dic3)⋊2C22, D6.21(C22×S3), C327D44C22, D6⋊S314C22, C3⋊D1215C22, C322Q813C22, (D4×C32)⋊12C22, C3⋊Dic3.22C23, C324Q810C22, Dic3.10(C22×S3), (C3×Dic3).14C23, (C3×S3×D4)⋊9C2, C4.32(C2×S32), (C2×S32)⋊4C22, (S3×C3⋊D4)⋊4C2, C22.4(C2×S32), (C4×C3⋊S3)⋊4C22, (S3×C2×C6)⋊10C22, C2.22(C22×S32), (C3×C3⋊D4)⋊4C22, (C2×C6).5(C22×S3), (C2×D6⋊S3)⋊15C2, (C2×C3⋊S3).25C23, (C2×C3⋊Dic3)⋊11C22, SmallGroup(288,961)

Series: Derived Chief Lower central Upper central

C1C3×C6 — D1212D6
C1C3C32C3×C6S3×C6C2×S32S3×C3⋊D4 — D1212D6
C32C3×C6 — D1212D6
C1C2D4

Generators and relations for D1212D6
 G = < a,b,c,d | a12=b2=c6=d2=1, bab=a-1, cac-1=a7, ad=da, bc=cb, dbd=a6b, dcd=c-1 >

Subgroups: 1338 in 352 conjugacy classes, 108 normal (18 characteristic)
C1, C2, C2 [×9], C3 [×2], C3, C4, C4 [×5], C22 [×2], C22 [×13], S3 [×9], C6 [×2], C6 [×13], C2×C4 [×9], D4, D4 [×17], Q8 [×2], C23 [×6], C32, Dic3 [×2], Dic3 [×9], C12 [×2], C12 [×3], D6 [×6], D6 [×11], C2×C6 [×4], C2×C6 [×12], C2×D4 [×9], C4○D4 [×6], C3×S3 [×6], C3⋊S3, C3×C6, C3×C6 [×2], Dic6 [×5], C4×S3 [×2], C4×S3 [×7], D12 [×2], D12 [×2], C2×Dic3 [×10], C3⋊D4 [×4], C3⋊D4 [×22], C2×C12 [×2], C3×D4 [×2], C3×D4 [×7], C22×S3 [×4], C22×S3 [×4], C22×C6 [×4], 2+ 1+4, C3×Dic3 [×2], C3⋊Dic3, C3⋊Dic3 [×2], C3×C12, S32 [×2], S3×C6 [×6], S3×C6 [×4], C2×C3⋊S3, C62 [×2], C4○D12 [×4], S3×D4 [×2], S3×D4 [×6], D42S3 [×9], C2×C3⋊D4 [×8], C6×D4 [×2], S3×Dic3 [×4], D6⋊S3, D6⋊S3 [×6], C3⋊D12 [×2], C322Q8, S3×C12 [×2], C3×D12 [×2], C3×C3⋊D4 [×4], C324Q8, C4×C3⋊S3, C2×C3⋊Dic3 [×2], C327D4 [×2], D4×C32, C2×S32 [×2], S3×C2×C6 [×4], D46D6 [×2], D125S3 [×2], D6.D6, D6⋊D6, D6.4D6 [×2], C2×D6⋊S3 [×2], S3×C3⋊D4 [×4], C3×S3×D4 [×2], C12.D6, D1212D6
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], C23 [×15], D6 [×14], C24, C22×S3 [×14], 2+ 1+4, S32, S3×C23 [×2], C2×S32 [×3], D46D6 [×2], C22×S32, D1212D6

Smallest permutation representation of D1212D6
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 45)(14 44)(15 43)(16 42)(17 41)(18 40)(19 39)(20 38)(21 37)(22 48)(23 47)(24 46)
(1 23 5 15 9 19)(2 18 6 22 10 14)(3 13 7 17 11 21)(4 20 8 24 12 16)(25 39 33 47 29 43)(26 46 34 42 30 38)(27 41 35 37 31 45)(28 48 36 44 32 40)
(1 46)(2 47)(3 48)(4 37)(5 38)(6 39)(7 40)(8 41)(9 42)(10 43)(11 44)(12 45)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(21 36)(22 25)(23 26)(24 27)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D···2I2J3A3B3C4A4B4C4D4E4F6A6B6C···6G6H6I6J6K6L6M6N6O6P6Q12A12B12C12D12E
order12222···22333444444666···666666666661212121212
size11226···618224266181818224···4666688121212124481212

42 irreducible representations

dim111111111222222444448
type+++++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2S3D6D6D6D6D62+ 1+4S32C2×S32C2×S32D46D6D1212D6
kernelD1212D6D125S3D6.D6D6⋊D6D6.4D6C2×D6⋊S3S3×C3⋊D4C3×S3×D4C12.D6S3×D4C4×S3D12C3⋊D4C3×D4C22×S3C32D4C4C22C3C1
# reps121122421222424111241

Matrix representation of D1212D6 in GL6(𝔽13)

1200000
0120000
00911126
0021176
006342
00103112
,
100000
010000
00121200
000100
00001212
000001
,
1210000
1200000
0012000
0001200
008010
000801
,
100000
1120000
0011900
004200
0000119
000042

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,9,2,6,10,0,0,11,11,3,3,0,0,12,7,4,11,0,0,6,6,2,2],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0,0,0,0,0,12,1],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,8,0,0,0,0,12,0,8,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,0,12,0,0,0,0,0,0,11,4,0,0,0,0,9,2,0,0,0,0,0,0,11,4,0,0,0,0,9,2] >;

D1212D6 in GAP, Magma, Sage, TeX

D_{12}\rtimes_{12}D_6
% in TeX

G:=Group("D12:12D6");
// GroupNames label

G:=SmallGroup(288,961);
// by ID

G=gap.SmallGroup(288,961);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,675,185,1356,9414]);
// 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=a^7,a*d=d*a,b*c=c*b,d*b*d=a^6*b,d*c*d=c^-1>;
// generators/relations

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