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G = D61Dic6order 288 = 25·32

1st semidirect product of D6 and Dic6 acting via Dic6/Dic3=C2

metabelian, supersoluble, monomial

Aliases: D61Dic6, C62.57C23, (S3×C6)⋊1Q8, D6⋊C4.8S3, C6.26(S3×Q8), (C2×Dic6)⋊4S3, (C6×Dic6)⋊9C2, (C2×C12).20D6, C6.140(S3×D4), C33(D63Q8), (C3×Dic3).6D4, C6.14(C2×Dic6), C2.15(S3×Dic6), C6.9(D42S3), C328(C22⋊Q8), D6⋊Dic3.13C2, (C2×Dic3).68D6, (C22×S3).36D6, Dic3⋊Dic334C2, (C6×C12).182C22, C6.33(Q83S3), C6.Dic610C2, Dic3.8(C3⋊D4), C2.16(D12⋊S3), C36(Dic3.D4), (C6×Dic3).36C22, (C2×C4).23S32, (C3×D6⋊C4).7C2, (C3×C6).95(C2×D4), C6.34(C2×C3⋊D4), C2.14(S3×C3⋊D4), (C3×C6).27(C2×Q8), (C2×S3×Dic3).2C2, C22.104(C2×S32), (S3×C2×C6).18C22, (C3×C6).34(C4○D4), (C2×C6).76(C22×S3), (C2×C3⋊Dic3).41C22, SmallGroup(288,535)

Series: Derived Chief Lower central Upper central

C1C62 — D61Dic6
C1C3C32C3×C6C62C6×Dic3C2×S3×Dic3 — D61Dic6
C32C62 — D61Dic6
C1C22C2×C4

Generators and relations for D61Dic6
 G = < a,b,c,d | a6=b2=c12=1, d2=c6, bab=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd-1=c-1 >

Subgroups: 554 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 [×9], C12 [×7], 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 [×2], C4×S3 [×2], C2×Dic3 [×3], C2×Dic3 [×8], C2×C12 [×2], C2×C12 [×4], C3×Q8 [×2], 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 [×5], C4⋊Dic3 [×2], D6⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, C2×Dic6, S3×C2×C4, C22×Dic3, C6×Q8, S3×Dic3 [×2], C3×Dic6 [×2], C6×Dic3 [×3], C2×C3⋊Dic3 [×2], C6×C12, S3×C2×C6, Dic3.D4, D63Q8, D6⋊Dic3, Dic3⋊Dic3 [×2], C3×D6⋊C4, C6.Dic6, C2×S3×Dic3, C6×Dic6, D61Dic6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], Q8 [×2], C23, D6 [×6], C2×D4, C2×Q8, C4○D4, Dic6 [×2], C3⋊D4 [×2], C22×S3 [×2], C22⋊Q8, S32, C2×Dic6, S3×D4, D42S3, S3×Q8, Q83S3, C2×C3⋊D4, C2×S32, Dic3.D4, D63Q8, S3×Dic6, D12⋊S3, S3×C3⋊D4, D61Dic6

Smallest permutation representation of D61Dic6
On 96 points
Generators in S96
(1 74 9 82 5 78)(2 75 10 83 6 79)(3 76 11 84 7 80)(4 77 12 73 8 81)(13 47 21 43 17 39)(14 48 22 44 18 40)(15 37 23 45 19 41)(16 38 24 46 20 42)(25 60 29 52 33 56)(26 49 30 53 34 57)(27 50 31 54 35 58)(28 51 32 55 36 59)(61 89 65 93 69 85)(62 90 66 94 70 86)(63 91 67 95 71 87)(64 92 68 96 72 88)
(1 33)(2 49)(3 35)(4 51)(5 25)(6 53)(7 27)(8 55)(9 29)(10 57)(11 31)(12 59)(13 64)(14 85)(15 66)(16 87)(17 68)(18 89)(19 70)(20 91)(21 72)(22 93)(23 62)(24 95)(26 75)(28 77)(30 79)(32 81)(34 83)(36 73)(37 90)(38 71)(39 92)(40 61)(41 94)(42 63)(43 96)(44 65)(45 86)(46 67)(47 88)(48 69)(50 84)(52 74)(54 76)(56 78)(58 80)(60 82)
(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 95 7 89)(2 94 8 88)(3 93 9 87)(4 92 10 86)(5 91 11 85)(6 90 12 96)(13 26 19 32)(14 25 20 31)(15 36 21 30)(16 35 22 29)(17 34 23 28)(18 33 24 27)(37 59 43 53)(38 58 44 52)(39 57 45 51)(40 56 46 50)(41 55 47 49)(42 54 48 60)(61 78 67 84)(62 77 68 83)(63 76 69 82)(64 75 70 81)(65 74 71 80)(66 73 72 79)

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

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

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

42 conjugacy classes

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

42 irreducible representations

dim11111112222222222444444444
type++++++++++-+++-++--++-
imageC1C2C2C2C2C2C2S3S3D4Q8D6D6D6C4○D4C3⋊D4Dic6S32S3×D4D42S3S3×Q8Q83S3C2×S32S3×Dic6D12⋊S3S3×C3⋊D4
kernelD61Dic6D6⋊Dic3Dic3⋊Dic3C3×D6⋊C4C6.Dic6C2×S3×Dic3C6×Dic6D6⋊C4C2×Dic6C3×Dic3S3×C6C2×Dic3C2×C12C22×S3C3×C6Dic3D6C2×C4C6C6C6C6C22C2C2C2
# reps11211111122321244111111222

Matrix representation of D61Dic6 in GL6(𝔽13)

1200000
0120000
000100
00121200
000010
000001
,
290000
4110000
0001200
0012000
000010
000001
,
010000
100000
0012000
0001200
0000610
000033
,
1200000
0120000
001000
000100
000050
000058

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],[2,4,0,0,0,0,9,11,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,6,3,0,0,0,0,10,3],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,5,0,0,0,0,0,8] >;

D61Dic6 in GAP, Magma, Sage, TeX

D_6\rtimes_1{\rm Dic}_6
% in TeX

G:=Group("D6:1Dic6");
// GroupNames label

G:=SmallGroup(288,535);
// by ID

G=gap.SmallGroup(288,535);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,422,135,100,1356,9414]);
// Polycyclic

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

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