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G = D62Dic6order 288 = 25·32

2nd semidirect product of D6 and Dic6 acting via Dic6/Dic3=C2

metabelian, supersoluble, monomial

Aliases: D62Dic6, Dic3.10D12, C62.63C23, (S3×C6)⋊2Q8, D6⋊C4.3S3, C6.17(S3×D4), C6.28(S3×Q8), Dic3⋊C45S3, (C2×C12).22D6, C2.20(S3×D12), C6.18(C2×D12), C32(C4.D12), D6⋊Dic3.6C2, (C6×C12).5C22, (C3×Dic3).8D4, C2.16(S3×Dic6), C6.16(C2×Dic6), Dic3⋊Dic36C2, C12⋊Dic34C2, (C2×Dic3).25D6, (C22×S3).39D6, C6.63(D42S3), C3210(C22⋊Q8), C31(Dic3.D4), C2.11(D6.4D6), (C6×Dic3).38C22, (C2×C4).27S32, (C3×D6⋊C4).1C2, (C3×C6).50(C2×D4), (C3×C6).30(C2×Q8), (C2×S3×Dic3).3C2, C22.108(C2×S32), (S3×C2×C6).22C22, (C3×Dic3⋊C4)⋊17C2, (C3×C6).66(C4○D4), (C2×C322Q8)⋊11C2, (C2×C6).82(C22×S3), (C2×C3⋊Dic3).47C22, SmallGroup(288,541)

Series: Derived Chief Lower central Upper central

C1C62 — D62Dic6
C1C3C32C3×C6C62C6×Dic3C2×S3×Dic3 — D62Dic6
C32C62 — D62Dic6
C1C22C2×C4

Generators and relations for D62Dic6
 G = < a,b,c,d | a6=b2=c12=1, d2=c6, bab=cac-1=a-1, ad=da, cbc-1=ab, bd=db, dcd-1=c-1 >

Subgroups: 586 in 161 conjugacy classes, 52 normal (44 characteristic)
C1, C2 [×3], C2 [×2], C3 [×2], C3, C4 [×7], C22, C22 [×4], S3 [×2], C6 [×6], C6 [×5], C2×C4, C2×C4 [×7], Q8 [×2], C23, C32, Dic3 [×2], Dic3 [×8], C12 [×8], D6 [×2], D6 [×2], C2×C6 [×2], C2×C6 [×5], C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, C3×S3 [×2], C3×C6 [×3], Dic6 [×4], C4×S3 [×2], C2×Dic3 [×3], C2×Dic3 [×8], C2×C12 [×2], C2×C12 [×4], C22×S3, C22×C6, C22⋊Q8, C3×Dic3 [×2], C3×Dic3 [×2], C3⋊Dic3 [×2], C3×C12, S3×C6 [×2], S3×C6 [×2], C62, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3 [×4], D6⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6 [×2], S3×C2×C4, C22×Dic3, S3×Dic3 [×2], C322Q8 [×2], C6×Dic3 [×3], C2×C3⋊Dic3 [×2], C6×C12, S3×C2×C6, Dic3.D4, C4.D12, D6⋊Dic3, Dic3⋊Dic3, C3×Dic3⋊C4, C3×D6⋊C4, C12⋊Dic3, C2×S3×Dic3, C2×C322Q8, D62Dic6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], Q8 [×2], C23, D6 [×6], C2×D4, C2×Q8, C4○D4, Dic6 [×2], D12 [×2], C22×S3 [×2], C22⋊Q8, S32, C2×Dic6, C2×D12, S3×D4, D42S3 [×2], S3×Q8, C2×S32, Dic3.D4, C4.D12, S3×Dic6, S3×D12, D6.4D6, D62Dic6

Smallest permutation representation of D62Dic6
On 96 points
Generators in S96
(1 36 9 32 5 28)(2 29 6 33 10 25)(3 26 11 34 7 30)(4 31 8 35 12 27)(13 71 17 63 21 67)(14 68 22 64 18 72)(15 61 19 65 23 69)(16 70 24 66 20 62)(37 96 41 88 45 92)(38 93 46 89 42 85)(39 86 43 90 47 94)(40 95 48 91 44 87)(49 80 57 76 53 84)(50 73 54 77 58 81)(51 82 59 78 55 74)(52 75 56 79 60 83)
(1 73)(2 51)(3 75)(4 53)(5 77)(6 55)(7 79)(8 57)(9 81)(10 59)(11 83)(12 49)(13 93)(14 39)(15 95)(16 41)(17 85)(18 43)(19 87)(20 45)(21 89)(22 47)(23 91)(24 37)(25 82)(26 52)(27 84)(28 54)(29 74)(30 56)(31 76)(32 58)(33 78)(34 60)(35 80)(36 50)(38 71)(40 61)(42 63)(44 65)(46 67)(48 69)(62 88)(64 90)(66 92)(68 94)(70 96)(72 86)
(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 23 7 17)(2 22 8 16)(3 21 9 15)(4 20 10 14)(5 19 11 13)(6 18 12 24)(25 68 31 62)(26 67 32 61)(27 66 33 72)(28 65 34 71)(29 64 35 70)(30 63 36 69)(37 55 43 49)(38 54 44 60)(39 53 45 59)(40 52 46 58)(41 51 47 57)(42 50 48 56)(73 91 79 85)(74 90 80 96)(75 89 81 95)(76 88 82 94)(77 87 83 93)(78 86 84 92)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D4E4F4G4H6A···6F6G6H6I6J6K12A···12H12I···12N
order122222333444444446···66666612···1212···12
size11116622446612121818362···244412124···412···12

42 irreducible representations

dim11111111222222222244444444
type+++++++++++-++++-++--+-+-
imageC1C2C2C2C2C2C2C2S3S3D4Q8D6D6D6C4○D4D12Dic6S32S3×D4D42S3S3×Q8C2×S32S3×Dic6S3×D12D6.4D6
kernelD62Dic6D6⋊Dic3Dic3⋊Dic3C3×Dic3⋊C4C3×D6⋊C4C12⋊Dic3C2×S3×Dic3C2×C322Q8Dic3⋊C4D6⋊C4C3×Dic3S3×C6C2×Dic3C2×C12C22×S3C3×C6Dic3D6C2×C4C6C6C6C22C2C2C2
# reps11111111112232124411211222

Matrix representation of D62Dic6 in GL6(𝔽13)

1200000
0120000
000100
00121200
000010
000001
,
1160000
620000
0001200
0012000
0000120
0000012
,
7110000
1160000
001000
00121200
0000103
0000107
,
100000
010000
001000
000100
0000112
000042

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

D62Dic6 in GAP, Magma, Sage, TeX

D_6\rtimes_2{\rm Dic}_6
% in TeX

G:=Group("D6:2Dic6");
// GroupNames label

G:=SmallGroup(288,541);
// by ID

G=gap.SmallGroup(288,541);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,422,135,142,1356,9414]);
// Polycyclic

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

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