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## G = Dic3⋊6Dic6order 288 = 25·32

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — Dic3⋊6Dic6
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — Dic32 — Dic3⋊6Dic6
 Lower central C32 — C3×C6 — Dic3⋊6Dic6
 Upper central C1 — C22 — C2×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, C3, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, Q8, C32, Dic3, Dic3, C12, C12, C2×C6, C2×C6, C42, C4⋊C4, C2×Q8, C3×C6, Dic6, C2×Dic3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C4×Q8, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, C62, C4×Dic3, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C3×C4⋊C4, C2×Dic6, C6×Dic3, C6×Dic3, C324Q8, C2×C3⋊Dic3, C6×C12, C4×Dic6, Dic6⋊C4, Dic32, C62.C22, Dic3×C12, C3×C4⋊Dic3, C2×C324Q8, Dic36Dic6
Quotients: C1, C2, C4, C22, S3, C2×C4, Q8, C23, D6, C22×C4, C2×Q8, C4○D4, Dic6, C4×S3, C22×S3, C4×Q8, S32, C2×Dic6, S3×C2×C4, C4○D12, D42S3, S3×Q8, C6.D6, 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 39 5 43 9 47)(2 40 6 44 10 48)(3 41 7 45 11 37)(4 42 8 46 12 38)(13 28 21 36 17 32)(14 29 22 25 18 33)(15 30 23 26 19 34)(16 31 24 27 20 35)(49 63 57 71 53 67)(50 64 58 72 54 68)(51 65 59 61 55 69)(52 66 60 62 56 70)(73 91 77 95 81 87)(74 92 78 96 82 88)(75 93 79 85 83 89)(76 94 80 86 84 90)
(1 34 43 23)(2 35 44 24)(3 36 45 13)(4 25 46 14)(5 26 47 15)(6 27 48 16)(7 28 37 17)(8 29 38 18)(9 30 39 19)(10 31 40 20)(11 32 41 21)(12 33 42 22)(49 74 71 96)(50 75 72 85)(51 76 61 86)(52 77 62 87)(53 78 63 88)(54 79 64 89)(55 80 65 90)(56 81 66 91)(57 82 67 92)(58 83 68 93)(59 84 69 94)(60 73 70 95)
(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 92 7 86)(2 91 8 85)(3 90 9 96)(4 89 10 95)(5 88 11 94)(6 87 12 93)(13 65 19 71)(14 64 20 70)(15 63 21 69)(16 62 22 68)(17 61 23 67)(18 72 24 66)(25 54 31 60)(26 53 32 59)(27 52 33 58)(28 51 34 57)(29 50 35 56)(30 49 36 55)(37 76 43 82)(38 75 44 81)(39 74 45 80)(40 73 46 79)(41 84 47 78)(42 83 48 77)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G ··· 4L 4M 4N 4O 4P 6A ··· 6F 6G 6H 6I 12A 12B 12C 12D 12E ··· 12J 12K ··· 12R 12S 12T 12U 12V order 1 2 2 2 3 3 3 4 4 4 4 4 4 4 ··· 4 4 4 4 4 6 ··· 6 6 6 6 12 12 12 12 12 ··· 12 12 ··· 12 12 12 12 12 size 1 1 1 1 2 2 4 2 2 3 3 3 3 6 ··· 6 18 18 18 18 2 ··· 2 4 4 4 2 2 2 2 4 ··· 4 6 ··· 6 12 12 12 12

54 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + - + + - + - - + + - - image C1 C2 C2 C2 C2 C2 C4 S3 S3 Q8 D6 D6 C4○D4 Dic6 C4×S3 C4○D12 S32 D4⋊2S3 S3×Q8 C6.D6 C2×S32 S3×Dic6 D12⋊5S3 kernel Dic3⋊6Dic6 Dic32 C62.C22 Dic3×C12 C3×C4⋊Dic3 C2×C32⋊4Q8 C32⋊4Q8 C4×Dic3 C4⋊Dic3 C3×Dic3 C2×Dic3 C2×C12 C3×C6 Dic3 C12 C6 C2×C4 C6 C6 C4 C22 C2 C2 # reps 1 2 2 1 1 1 8 1 1 2 4 2 2 4 8 4 1 1 1 2 1 2 2

Matrix representation of Dic36Dic6 in GL6(𝔽13)

 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 12 0 0 0 0 1 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 3 0 0 0 0 8 1 0 0 0 0 0 0 12 12 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 1 0 0 0 0 11 12 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 0 1 0 0 0 0 1 0

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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