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## G = D6⋊3Dic6order 288 = 25·32

### 3rd semidirect product of D6 and Dic6 acting via Dic6/Dic3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — D6⋊3Dic6
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×S3×Dic3 — D6⋊3Dic6
 Lower central C32 — C62 — D6⋊3Dic6
 Upper central C1 — C22 — C2×C4

Generators and relations for D63Dic6
G = < a,b,c,d | a6=b2=c12=1, d2=c6, bab=dad-1=a-1, ac=ca, cbc-1=a3b, dbd-1=a4b, dcd-1=c-1 >

Subgroups: 610 in 167 conjugacy classes, 50 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 [×13], 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 [×8], C4×S3 [×2], C2×Dic3 [×3], C2×Dic3 [×8], C2×C12 [×2], C2×C12 [×4], C22×S3, C22×C6, C22⋊Q8, C3×Dic3 [×3], C3⋊Dic3 [×2], C3⋊Dic3, C3×C12, S3×C6 [×2], S3×C6 [×2], C62, Dic3⋊C4, Dic3⋊C4 [×3], C4⋊Dic3, D6⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6 [×3], S3×C2×C4, C22×Dic3, S3×Dic3 [×2], C6×Dic3 [×3], C324Q8 [×2], C2×C3⋊Dic3 [×2], C6×C12, S3×C2×C6, Dic3.D4, D6⋊Q8, D6⋊Dic3, Dic3⋊Dic3, C62.C22, C3×Dic3⋊C4, C3×D6⋊C4, C2×S3×Dic3, C2×C324Q8, D63Dic6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], Q8 [×2], C23, D6 [×6], C2×D4, C2×Q8, C4○D4, Dic6 [×2], C22×S3 [×2], C22⋊Q8, S32, C2×Dic6, C4○D12, S3×D4 [×2], D42S3, S3×Q8, C2×S32, Dic3.D4, D6⋊Q8, S3×Dic6, D125S3, Dic3⋊D6, D63Dic6

Smallest permutation representation of D63Dic6
On 96 points
Generators in S96
(1 46 9 42 5 38)(2 47 10 43 6 39)(3 48 11 44 7 40)(4 37 12 45 8 41)(13 26 21 34 17 30)(14 27 22 35 18 31)(15 28 23 36 19 32)(16 29 24 25 20 33)(49 75 53 79 57 83)(50 76 54 80 58 84)(51 77 55 81 59 73)(52 78 56 82 60 74)(61 92 65 96 69 88)(62 93 66 85 70 89)(63 94 67 86 71 90)(64 95 68 87 72 91)
(1 66)(2 90)(3 68)(4 92)(5 70)(6 94)(7 72)(8 96)(9 62)(10 86)(11 64)(12 88)(13 74)(14 57)(15 76)(16 59)(17 78)(18 49)(19 80)(20 51)(21 82)(22 53)(23 84)(24 55)(25 77)(26 60)(27 79)(28 50)(29 81)(30 52)(31 83)(32 54)(33 73)(34 56)(35 75)(36 58)(37 61)(38 85)(39 63)(40 87)(41 65)(42 89)(43 67)(44 91)(45 69)(46 93)(47 71)(48 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 28 7 34)(2 27 8 33)(3 26 9 32)(4 25 10 31)(5 36 11 30)(6 35 12 29)(13 42 19 48)(14 41 20 47)(15 40 21 46)(16 39 22 45)(17 38 23 44)(18 37 24 43)(49 65 55 71)(50 64 56 70)(51 63 57 69)(52 62 58 68)(53 61 59 67)(54 72 60 66)(73 94 79 88)(74 93 80 87)(75 92 81 86)(76 91 82 85)(77 90 83 96)(78 89 84 95)

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + - + + + - + + - - + - - + image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D4 Q8 D6 D6 D6 C4○D4 Dic6 C4○D12 S32 S3×D4 D4⋊2S3 S3×Q8 C2×S32 S3×Dic6 D12⋊5S3 Dic3⋊D6 kernel D6⋊3Dic6 D6⋊Dic3 Dic3⋊Dic3 C62.C22 C3×Dic3⋊C4 C3×D6⋊C4 C2×S3×Dic3 C2×C32⋊4Q8 Dic3⋊C4 D6⋊C4 C3⋊Dic3 S3×C6 C2×Dic3 C2×C12 C22×S3 C3×C6 D6 C6 C2×C4 C6 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 1 1 2 2 3 2 1 2 4 4 1 2 1 1 1 2 2 2

Matrix representation of D63Dic6 in GL6(𝔽13)

 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 1 12 0 0 0 0 1 0
,
 11 9 0 0 0 0 4 2 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 7 3 0 0 0 0 10 6
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 1 12 0 0 0 0 0 0 10 6 0 0 0 0 7 3
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 2 2 0 0 0 0 4 11

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,12,0],[11,4,0,0,0,0,9,2,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,7,10,0,0,0,0,3,6],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,10,7,0,0,0,0,6,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,4,0,0,0,0,2,11] >;

D63Dic6 in GAP, Magma, Sage, TeX

D_6\rtimes_3{\rm Dic}_6
% in TeX

G:=Group("D6:3Dic6");
// GroupNames label

G:=SmallGroup(288,544);
// by ID

G=gap.SmallGroup(288,544);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,590,219,58,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^2=c^12=1,d^2=c^6,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^3*b,d*b*d^-1=a^4*b,d*c*d^-1=c^-1>;
// generators/relations

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