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G = Dic36Dic6order 288 = 25·32

2nd semidirect product of Dic3 and Dic6 acting through Inn(Dic3)

metabelian, supersoluble, monomial

Aliases: Dic36Dic6, C62.14C23, C326(C4×Q8), C33(C4×Dic6), Dic32.9C2, C6.17(S3×Q8), C12.35(C4×S3), (C3×Dic3)⋊7Q8, C6.5(C2×Dic6), C2.3(S3×Dic6), (C2×C12).128D6, C4⋊Dic3.14S3, (C4×Dic3).2S3, C324Q810C4, C6.37(C4○D12), (C6×C12).88C22, (Dic3×C12).6C2, C4.5(C6.D6), C32(Dic6⋊C4), C2.2(D125S3), C6.14(D42S3), (C2×Dic3).109D6, C62.C22.11C2, (C6×Dic3).150C22, (C2×C4).71S32, C6.29(S3×C2×C4), C22.21(C2×S32), (C3×C6).12(C2×Q8), (C3×C12).63(C2×C4), (C3×C6).5(C4○D4), C2.7(C2×C6.D6), (C3×C4⋊Dic3).11C2, C3⋊Dic3.21(C2×C4), (C2×C6).33(C22×S3), (C3×C6).49(C22×C4), (C2×C324Q8).12C2, (C2×C3⋊Dic3).15C22, SmallGroup(288,492)

Series: Derived Chief Lower central Upper central

C1C3×C6 — Dic36Dic6
C1C3C32C3×C6C62C6×Dic3Dic32 — Dic36Dic6
C32C3×C6 — Dic36Dic6
C1C22C2×C4

Generators and relations for Dic36Dic6
 G = < a,b,c,d | a6=c12=1, b2=a3, d2=c6, bab-1=dad-1=a-1, ac=ca, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 482 in 159 conjugacy classes, 64 normal (34 characteristic)
C1, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×9], C22, C6 [×6], C6 [×3], C2×C4, C2×C4 [×6], Q8 [×4], C32, Dic3 [×2], Dic3 [×15], C12 [×4], C12 [×7], C2×C6 [×2], C2×C6, C42 [×3], C4⋊C4 [×3], C2×Q8, C3×C6 [×3], Dic6 [×12], C2×Dic3 [×2], C2×Dic3 [×2], C2×Dic3 [×6], C2×C12 [×2], C2×C12 [×5], C4×Q8, C3×Dic3 [×2], C3×Dic3 [×3], C3⋊Dic3 [×4], C3×C12 [×2], C62, C4×Dic3, C4×Dic3 [×4], Dic3⋊C4 [×4], C4⋊Dic3, C4×C12, C3×C4⋊C4, C2×Dic6 [×3], C6×Dic3 [×2], C6×Dic3 [×2], C324Q8 [×4], C2×C3⋊Dic3 [×2], C6×C12, C4×Dic6, Dic6⋊C4, Dic32 [×2], C62.C22 [×2], Dic3×C12, C3×C4⋊Dic3, C2×C324Q8, Dic36Dic6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], Q8 [×2], C23, D6 [×6], C22×C4, C2×Q8, C4○D4, Dic6 [×2], C4×S3 [×4], C22×S3 [×2], C4×Q8, S32, C2×Dic6, S3×C2×C4 [×2], C4○D12, D42S3, S3×Q8, C6.D6 [×2], C2×S32, C4×Dic6, Dic6⋊C4, S3×Dic6, D125S3, C2×C6.D6, Dic36Dic6

Smallest permutation representation of Dic36Dic6
On 96 points
Generators in S96
(1 72 5 64 9 68)(2 61 6 65 10 69)(3 62 7 66 11 70)(4 63 8 67 12 71)(13 28 21 36 17 32)(14 29 22 25 18 33)(15 30 23 26 19 34)(16 31 24 27 20 35)(37 80 45 76 41 84)(38 81 46 77 42 73)(39 82 47 78 43 74)(40 83 48 79 44 75)(49 92 53 96 57 88)(50 93 54 85 58 89)(51 94 55 86 59 90)(52 95 56 87 60 91)
(1 81 64 42)(2 82 65 43)(3 83 66 44)(4 84 67 45)(5 73 68 46)(6 74 69 47)(7 75 70 48)(8 76 71 37)(9 77 72 38)(10 78 61 39)(11 79 62 40)(12 80 63 41)(13 58 36 93)(14 59 25 94)(15 60 26 95)(16 49 27 96)(17 50 28 85)(18 51 29 86)(19 52 30 87)(20 53 31 88)(21 54 32 89)(22 55 33 90)(23 56 34 91)(24 57 35 92)
(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 56 7 50)(2 55 8 49)(3 54 9 60)(4 53 10 59)(5 52 11 58)(6 51 12 57)(13 46 19 40)(14 45 20 39)(15 44 21 38)(16 43 22 37)(17 42 23 48)(18 41 24 47)(25 84 31 78)(26 83 32 77)(27 82 33 76)(28 81 34 75)(29 80 35 74)(30 79 36 73)(61 94 67 88)(62 93 68 87)(63 92 69 86)(64 91 70 85)(65 90 71 96)(66 89 72 95)

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

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

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

54 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D4E4F4G···4L4M4N4O4P6A···6F6G6H6I12A12B12C12D12E···12J12K···12R12S12T12U12V
order12223334444444···444446···66661212121212···1212···1212121212
size11112242233336···6181818182···244422224···46···612121212

54 irreducible representations

dim11111112222222224444444
type++++++++-++-+--++--
imageC1C2C2C2C2C2C4S3S3Q8D6D6C4○D4Dic6C4×S3C4○D12S32D42S3S3×Q8C6.D6C2×S32S3×Dic6D125S3
kernelDic36Dic6Dic32C62.C22Dic3×C12C3×C4⋊Dic3C2×C324Q8C324Q8C4×Dic3C4⋊Dic3C3×Dic3C2×Dic3C2×C12C3×C6Dic3C12C6C2×C4C6C6C4C22C2C2
# reps12211181124224841112122

Matrix representation of Dic36Dic6 in GL6(𝔽13)

100000
010000
0012000
0001200
0000012
0000112
,
100000
010000
005000
000500
000001
000010
,
1230000
810000
00121200
001000
000010
000001
,
110000
11120000
001000
00121200
000001
000010

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,8,0,0,0,0,3,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,11,0,0,0,0,1,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

Dic36Dic6 in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes_6{\rm Dic}_6
% in TeX

G:=Group("Dic3:6Dic6");
// GroupNames label

G:=SmallGroup(288,492);
// by ID

G=gap.SmallGroup(288,492);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,64,590,142,1356,9414]);
// Polycyclic

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

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