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## G = Dic3.Dic6order 288 = 25·32

### 1st non-split extension by Dic3 of Dic6 acting via Dic6/Dic3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — Dic3.Dic6
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — Dic32 — Dic3.Dic6
 Lower central C32 — C62 — Dic3.Dic6
 Upper central C1 — C22 — C2×C4

Generators and relations for Dic3.Dic6
G = < a,b,c,d | a6=c12=1, b2=a3, d2=c6, bab-1=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd-1=a3c-1 >

Subgroups: 394 in 125 conjugacy classes, 48 normal (44 characteristic)
C1, C2, C3, C3, C4, C22, C6, C6, C2×C4, C2×C4, C32, Dic3, Dic3, C12, C2×C6, C2×C6, C42, C4⋊C4, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C42.C2, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, C62, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C6×Dic3, C2×C3⋊Dic3, C6×C12, Dic3.Q8, C4.Dic6, Dic32, Dic3⋊Dic3, C62.C22, C3×Dic3⋊C4, C12⋊Dic3, Dic3.Dic6
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C4○D4, Dic6, C22×S3, C42.C2, S32, C2×Dic6, C4○D12, D42S3, S3×Q8, Q83S3, C2×S32, Dic3.Q8, C4.Dic6, S3×Dic6, D6.6D6, D6.4D6, Dic3.Dic6

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

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A ··· 6F 6G 6H 6I 12A ··· 12H 12I ··· 12P order 1 2 2 2 3 3 3 4 4 4 4 4 4 4 4 4 4 6 ··· 6 6 6 6 12 ··· 12 12 ··· 12 size 1 1 1 1 2 2 4 4 6 6 6 6 12 12 18 18 36 2 ··· 2 4 4 4 4 ··· 4 12 ··· 12

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + - + + - + - - + + - + - image C1 C2 C2 C2 C2 C2 S3 Q8 D6 D6 C4○D4 Dic6 C4○D12 S32 D4⋊2S3 S3×Q8 Q8⋊3S3 C2×S32 S3×Dic6 D6.6D6 D6.4D6 kernel Dic3.Dic6 Dic32 Dic3⋊Dic3 C62.C22 C3×Dic3⋊C4 C12⋊Dic3 Dic3⋊C4 C3×Dic3 C2×Dic3 C2×C12 C3×C6 Dic3 C6 C2×C4 C6 C6 C6 C22 C2 C2 C2 # reps 1 1 2 1 2 1 2 2 4 2 4 4 4 1 2 1 1 1 2 2 2

Matrix representation of Dic3.Dic6 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 10 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 10 0 0 0 0 9 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 8 11 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 8 0 0 0 0 0 0 12 12 0 0 0 0 1 0
,
 1 3 0 0 0 0 8 12 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 12 0 0 0 0 0 1 1

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,9,0,0,0,0,10,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[8,0,0,0,0,0,11,5,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[1,8,0,0,0,0,3,12,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,1,0,0,0,0,0,1] >;

Dic3.Dic6 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(288,493);
// by ID

G=gap.SmallGroup(288,493);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,64,590,135,100,1356,9414]);
// Polycyclic

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

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