Copied to
clipboard

G = Dic6⋊Dic3order 288 = 25·32

3rd semidirect product of Dic6 and Dic3 acting via Dic3/C6=C2

metabelian, supersoluble, monomial

Aliases: C12.16D12, C62.20D4, Dic63Dic3, (C3×C6).3Q16, C12.16(C4×S3), (C3×Dic6)⋊1C4, (C2×C12).78D6, (C3×C12).35D4, C4⋊Dic3.8S3, C4.7(S3×Dic3), C6.36(D6⋊C4), C6.5(D4.S3), (C2×Dic6).6S3, (C6×Dic6).2C2, C12.8(C2×Dic3), (C3×C6).10SD16, C6.4(C3⋊Q16), C33(C6.SD16), C32(Q82Dic3), C12.31(C3⋊D4), C2.7(D6⋊Dic3), (C6×C12).27C22, C6.5(Q82S3), C324(Q8⋊C4), C4.22(C3⋊D12), C6.6(C6.D4), C2.1(C322Q16), C2.2(Dic6⋊S3), C22.8(D6⋊S3), (C2×C4).100S32, (C3×C12).25(C2×C4), (C3×C4⋊Dic3).1C2, (C2×C6).49(C3⋊D4), (C2×C324C8).1C2, (C3×C6).32(C22⋊C4), SmallGroup(288,213)

Series: Derived Chief Lower central Upper central

C1C3×C12 — Dic6⋊Dic3
C1C3C32C3×C6C62C6×C12C6×Dic6 — Dic6⋊Dic3
C32C3×C6C3×C12 — Dic6⋊Dic3
C1C22C2×C4

Generators and relations for Dic6⋊Dic3
 G = < a,b,c,d | a12=c6=1, b2=a6, d2=c3, bab-1=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a9b, dcd-1=c-1 >

Subgroups: 274 in 95 conjugacy classes, 42 normal (38 characteristic)
C1, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×3], C22, C6 [×6], C6 [×3], C8, C2×C4, C2×C4 [×2], Q8 [×3], C32, Dic3 [×3], C12 [×4], C12 [×5], C2×C6 [×2], C2×C6, C4⋊C4, C2×C8, C2×Q8, C3×C6 [×3], C3⋊C8 [×4], Dic6 [×2], Dic6, C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×3], C3×Q8 [×3], Q8⋊C4, C3×Dic3 [×3], C3×C12 [×2], C62, C2×C3⋊C8 [×3], C4⋊Dic3, C3×C4⋊C4, C2×Dic6, C6×Q8, C324C8, C3×Dic6 [×2], C3×Dic6, C6×Dic3 [×2], C6×C12, C6.SD16, Q82Dic3, C3×C4⋊Dic3, C2×C324C8, C6×Dic6, Dic6⋊Dic3
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, D4 [×2], Dic3 [×2], D6 [×2], C22⋊C4, SD16, Q16, C4×S3, D12, C2×Dic3, C3⋊D4 [×3], Q8⋊C4, S32, D6⋊C4, D4.S3, Q82S3, C3⋊Q16 [×2], C6.D4, S3×Dic3, D6⋊S3, C3⋊D12, C6.SD16, Q82Dic3, Dic6⋊S3, C322Q16, D6⋊Dic3, Dic6⋊Dic3

Smallest permutation representation of Dic6⋊Dic3
On 96 points
Generators in S96
(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)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)
(1 61 7 67)(2 72 8 66)(3 71 9 65)(4 70 10 64)(5 69 11 63)(6 68 12 62)(13 51 19 57)(14 50 20 56)(15 49 21 55)(16 60 22 54)(17 59 23 53)(18 58 24 52)(25 86 31 92)(26 85 32 91)(27 96 33 90)(28 95 34 89)(29 94 35 88)(30 93 36 87)(37 83 43 77)(38 82 44 76)(39 81 45 75)(40 80 46 74)(41 79 47 73)(42 78 48 84)
(1 14 5 18 9 22)(2 15 6 19 10 23)(3 16 7 20 11 24)(4 17 8 21 12 13)(25 38 33 46 29 42)(26 39 34 47 30 43)(27 40 35 48 31 44)(28 41 36 37 32 45)(49 68 57 64 53 72)(50 69 58 65 54 61)(51 70 59 66 55 62)(52 71 60 67 56 63)(73 93 77 85 81 89)(74 94 78 86 82 90)(75 95 79 87 83 91)(76 96 80 88 84 92)
(1 44 18 35)(2 39 19 30)(3 46 20 25)(4 41 21 32)(5 48 22 27)(6 43 23 34)(7 38 24 29)(8 45 13 36)(9 40 14 31)(10 47 15 26)(11 42 16 33)(12 37 17 28)(49 88 64 76)(50 95 65 83)(51 90 66 78)(52 85 67 73)(53 92 68 80)(54 87 69 75)(55 94 70 82)(56 89 71 77)(57 96 72 84)(58 91 61 79)(59 86 62 74)(60 93 63 81)

G:=sub<Sym(96)| (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,61,7,67)(2,72,8,66)(3,71,9,65)(4,70,10,64)(5,69,11,63)(6,68,12,62)(13,51,19,57)(14,50,20,56)(15,49,21,55)(16,60,22,54)(17,59,23,53)(18,58,24,52)(25,86,31,92)(26,85,32,91)(27,96,33,90)(28,95,34,89)(29,94,35,88)(30,93,36,87)(37,83,43,77)(38,82,44,76)(39,81,45,75)(40,80,46,74)(41,79,47,73)(42,78,48,84), (1,14,5,18,9,22)(2,15,6,19,10,23)(3,16,7,20,11,24)(4,17,8,21,12,13)(25,38,33,46,29,42)(26,39,34,47,30,43)(27,40,35,48,31,44)(28,41,36,37,32,45)(49,68,57,64,53,72)(50,69,58,65,54,61)(51,70,59,66,55,62)(52,71,60,67,56,63)(73,93,77,85,81,89)(74,94,78,86,82,90)(75,95,79,87,83,91)(76,96,80,88,84,92), (1,44,18,35)(2,39,19,30)(3,46,20,25)(4,41,21,32)(5,48,22,27)(6,43,23,34)(7,38,24,29)(8,45,13,36)(9,40,14,31)(10,47,15,26)(11,42,16,33)(12,37,17,28)(49,88,64,76)(50,95,65,83)(51,90,66,78)(52,85,67,73)(53,92,68,80)(54,87,69,75)(55,94,70,82)(56,89,71,77)(57,96,72,84)(58,91,61,79)(59,86,62,74)(60,93,63,81)>;

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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,61,7,67)(2,72,8,66)(3,71,9,65)(4,70,10,64)(5,69,11,63)(6,68,12,62)(13,51,19,57)(14,50,20,56)(15,49,21,55)(16,60,22,54)(17,59,23,53)(18,58,24,52)(25,86,31,92)(26,85,32,91)(27,96,33,90)(28,95,34,89)(29,94,35,88)(30,93,36,87)(37,83,43,77)(38,82,44,76)(39,81,45,75)(40,80,46,74)(41,79,47,73)(42,78,48,84), (1,14,5,18,9,22)(2,15,6,19,10,23)(3,16,7,20,11,24)(4,17,8,21,12,13)(25,38,33,46,29,42)(26,39,34,47,30,43)(27,40,35,48,31,44)(28,41,36,37,32,45)(49,68,57,64,53,72)(50,69,58,65,54,61)(51,70,59,66,55,62)(52,71,60,67,56,63)(73,93,77,85,81,89)(74,94,78,86,82,90)(75,95,79,87,83,91)(76,96,80,88,84,92), (1,44,18,35)(2,39,19,30)(3,46,20,25)(4,41,21,32)(5,48,22,27)(6,43,23,34)(7,38,24,29)(8,45,13,36)(9,40,14,31)(10,47,15,26)(11,42,16,33)(12,37,17,28)(49,88,64,76)(50,95,65,83)(51,90,66,78)(52,85,67,73)(53,92,68,80)(54,87,69,75)(55,94,70,82)(56,89,71,77)(57,96,72,84)(58,91,61,79)(59,86,62,74)(60,93,63,81) );

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),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,61,7,67),(2,72,8,66),(3,71,9,65),(4,70,10,64),(5,69,11,63),(6,68,12,62),(13,51,19,57),(14,50,20,56),(15,49,21,55),(16,60,22,54),(17,59,23,53),(18,58,24,52),(25,86,31,92),(26,85,32,91),(27,96,33,90),(28,95,34,89),(29,94,35,88),(30,93,36,87),(37,83,43,77),(38,82,44,76),(39,81,45,75),(40,80,46,74),(41,79,47,73),(42,78,48,84)], [(1,14,5,18,9,22),(2,15,6,19,10,23),(3,16,7,20,11,24),(4,17,8,21,12,13),(25,38,33,46,29,42),(26,39,34,47,30,43),(27,40,35,48,31,44),(28,41,36,37,32,45),(49,68,57,64,53,72),(50,69,58,65,54,61),(51,70,59,66,55,62),(52,71,60,67,56,63),(73,93,77,85,81,89),(74,94,78,86,82,90),(75,95,79,87,83,91),(76,96,80,88,84,92)], [(1,44,18,35),(2,39,19,30),(3,46,20,25),(4,41,21,32),(5,48,22,27),(6,43,23,34),(7,38,24,29),(8,45,13,36),(9,40,14,31),(10,47,15,26),(11,42,16,33),(12,37,17,28),(49,88,64,76),(50,95,65,83),(51,90,66,78),(52,85,67,73),(53,92,68,80),(54,87,69,75),(55,94,70,82),(56,89,71,77),(57,96,72,84),(58,91,61,79),(59,86,62,74),(60,93,63,81)])

42 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D4E4F6A···6F6G6H6I8A8B8C8D12A···12H12I···12P
order12223334444446···6666888812···1212···12
size111122422121212122···2444181818184···412···12

42 irreducible representations

dim11111222222222222444444444
type++++++++-+-++-+--+-
imageC1C2C2C2C4S3S3D4D4Dic3D6SD16Q16C4×S3D12C3⋊D4C3⋊D4S32D4.S3Q82S3C3⋊Q16S3×Dic3C3⋊D12D6⋊S3Dic6⋊S3C322Q16
kernelDic6⋊Dic3C3×C4⋊Dic3C2×C324C8C6×Dic6C3×Dic6C4⋊Dic3C2×Dic6C3×C12C62Dic6C2×C12C3×C6C3×C6C12C12C12C2×C6C2×C4C6C6C6C4C4C22C2C2
# reps11114111122222224111211122

Matrix representation of Dic6⋊Dic3 in GL6(𝔽73)

010000
7200000
001000
000100
000011
0000720
,
6670000
67670000
0072000
0007200
0000072
0000720
,
100000
010000
000100
0072100
000010
000001
,
7200000
010000
0004600
0046000
00004313
00006030

G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,0,0,1,0],[6,67,0,0,0,0,67,67,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,72,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,46,0,0,0,0,46,0,0,0,0,0,0,0,43,60,0,0,0,0,13,30] >;

Dic6⋊Dic3 in GAP, Magma, Sage, TeX

{\rm Dic}_6\rtimes {\rm Dic}_3
% in TeX

G:=Group("Dic6:Dic3");
// GroupNames label

G:=SmallGroup(288,213);
// by ID

G=gap.SmallGroup(288,213);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,141,36,675,346,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^12=c^6=1,b^2=a^6,d^2=c^3,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^9*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽