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## G = D6⋊6Dic6order 288 = 25·32

### 2nd semidirect product of D6 and Dic6 acting via Dic6/C12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — D6⋊6Dic6
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — D6⋊Dic3 — D6⋊6Dic6
 Lower central C32 — C62 — D6⋊6Dic6
 Upper central C1 — C22 — C2×C4

Generators and relations for D66Dic6
G = < a,b,c,d | a6=b2=c12=1, d2=c6, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c-1 >

Subgroups: 522 in 161 conjugacy classes, 56 normal (34 characteristic)
C1, C2 [×3], C2 [×2], C3 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×4], S3 [×2], C6 [×6], C6 [×5], C2×C4, C2×C4 [×7], Q8 [×2], C23, C32, Dic3 [×9], C12 [×4], C12 [×5], 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, C2×Dic3 [×2], C2×Dic3 [×6], C2×C12 [×2], C2×C12 [×6], C3×Q8 [×2], C22×S3, C22×C6, C22⋊Q8, C3×Dic3 [×3], C3⋊Dic3 [×2], C3×C12 [×2], S3×C6 [×2], S3×C6 [×2], C62, Dic3⋊C4 [×4], C4⋊Dic3 [×3], D6⋊C4 [×2], C6.D4 [×2], C2×Dic6, S3×C2×C4, C22×C12, C6×Q8, C3×Dic6 [×2], S3×C12 [×2], C6×Dic3, C6×Dic3 [×2], C2×C3⋊Dic3 [×2], C6×C12, S3×C2×C6, C12.48D4, D63Q8, D6⋊Dic3 [×2], C62.C22 [×2], C12⋊Dic3, C6×Dic6, S3×C2×C12, D66Dic6
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 [×4], C22×S3 [×2], C22⋊Q8, S32, C2×Dic6, C4○D12, S3×Q8, Q83S3, C2×C3⋊D4 [×2], D6⋊S3 [×2], C2×S32, C12.48D4, D63Q8, S3×Dic6, D6.6D6, C2×D6⋊S3, D66Dic6

Smallest permutation representation of D66Dic6
On 96 points
Generators in S96
(1 21 9 17 5 13)(2 22 10 18 6 14)(3 23 11 19 7 15)(4 24 12 20 8 16)(25 83 29 75 33 79)(26 84 30 76 34 80)(27 73 31 77 35 81)(28 74 32 78 36 82)(37 92 41 96 45 88)(38 93 42 85 46 89)(39 94 43 86 47 90)(40 95 44 87 48 91)(49 61 57 69 53 65)(50 62 58 70 54 66)(51 63 59 71 55 67)(52 64 60 72 56 68)
(1 47)(2 48)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 44)(11 45)(12 46)(13 90)(14 91)(15 92)(16 93)(17 94)(18 95)(19 96)(20 85)(21 86)(22 87)(23 88)(24 89)(25 56)(26 57)(27 58)(28 59)(29 60)(30 49)(31 50)(32 51)(33 52)(34 53)(35 54)(36 55)(61 84)(62 73)(63 74)(64 75)(65 76)(66 77)(67 78)(68 79)(69 80)(70 81)(71 82)(72 83)
(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 82 7 76)(2 81 8 75)(3 80 9 74)(4 79 10 73)(5 78 11 84)(6 77 12 83)(13 36 19 30)(14 35 20 29)(15 34 21 28)(16 33 22 27)(17 32 23 26)(18 31 24 25)(37 49 43 55)(38 60 44 54)(39 59 45 53)(40 58 46 52)(41 57 47 51)(42 56 48 50)(61 86 67 92)(62 85 68 91)(63 96 69 90)(64 95 70 89)(65 94 71 88)(66 93 72 87)

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

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

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

48 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 6L 6M 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N 12O 12P 12Q 12R 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 6 6 12 12 12 12 12 ··· 12 12 12 12 12 12 12 12 12 size 1 1 1 1 6 6 2 2 4 2 2 6 6 12 12 36 36 2 ··· 2 4 4 4 6 6 6 6 2 2 2 2 4 ··· 4 6 6 6 6 12 12 12 12

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + + - + + + - + - + - + - + image C1 C2 C2 C2 C2 C2 S3 S3 D4 Q8 D6 D6 D6 C4○D4 C3⋊D4 Dic6 C4○D12 S32 S3×Q8 Q8⋊3S3 D6⋊S3 C2×S32 S3×Dic6 D6.6D6 kernel D6⋊6Dic6 D6⋊Dic3 C62.C22 C12⋊Dic3 C6×Dic6 S3×C2×C12 C2×Dic6 S3×C2×C4 C3×C12 S3×C6 C2×Dic3 C2×C12 C22×S3 C3×C6 C12 D6 C6 C2×C4 C6 C6 C4 C22 C2 C2 # reps 1 2 2 1 1 1 1 1 2 2 3 2 1 2 8 4 4 1 1 1 2 1 2 2

Matrix representation of D66Dic6 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 0 12 0 0 0 0 1 1
,
 12 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 6 3 0 0 0 0 10 7
,
 5 0 0 0 0 0 0 8 0 0 0 0 0 0 0 1 0 0 0 0 12 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 12 0 0 0 0 0 0 0 11 9 0 0 0 0 4 2

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,1],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,6,10,0,0,0,0,3,7],[5,0,0,0,0,0,0,8,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,11,4,0,0,0,0,9,2] >;

D66Dic6 in GAP, Magma, Sage, TeX

D_6\rtimes_6{\rm Dic}_6
% in TeX

G:=Group("D6:6Dic6");
// GroupNames label

G:=SmallGroup(288,504);
// by ID

G=gap.SmallGroup(288,504);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,120,219,100,1356,9414]);
// 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,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations

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