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## G = D6⋊Dic6order 288 = 25·32

### 1st semidirect product of D6 and Dic6 acting via Dic6/C12=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for D6⋊Dic6
G = < a,b,c,d | a6=b2=c12=1, d2=c6, bab=cac-1=dad-1=a-1, cbc-1=a4b, dbd-1=ab, dcd-1=c-1 >

Subgroups: 554 in 161 conjugacy classes, 52 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 [×2], Dic3 [×9], 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 [×4], C4×S3 [×2], C2×Dic3 [×3], C2×Dic3 [×6], C2×C12 [×2], C2×C12 [×6], C22×S3, C22×C6, C22⋊Q8, C3×Dic3 [×2], C3×Dic3 [×2], C3⋊Dic3 [×2], C3×C12, S3×C6 [×2], S3×C6 [×2], C62, Dic3⋊C4, Dic3⋊C4 [×4], C4⋊Dic3, D6⋊C4 [×2], C6.D4 [×2], C3×C4⋊C4, C2×Dic6 [×2], S3×C2×C4, C22×C12, C322Q8 [×2], S3×C12 [×2], C6×Dic3 [×3], C2×C3⋊Dic3 [×2], C6×C12, S3×C2×C6, D6⋊Q8, C12.48D4, D6⋊Dic3 [×2], Dic3⋊Dic3, C3×Dic3⋊C4, C6.Dic6, C2×C322Q8, S3×C2×C12, D6⋊Dic6
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 [×2], C22×S3 [×2], C22⋊Q8, S32, C2×Dic6, C4○D12 [×2], S3×D4, S3×Q8, C2×C3⋊D4, C2×S32, D6⋊Q8, C12.48D4, S3×Dic6, D6.D6, S3×C3⋊D4, D6⋊Dic6

Smallest permutation representation of D6⋊Dic6
On 96 points
Generators in S96
(1 28 9 36 5 32)(2 33 6 25 10 29)(3 30 11 26 7 34)(4 35 8 27 12 31)(13 62 21 70 17 66)(14 67 18 71 22 63)(15 64 23 72 19 68)(16 69 20 61 24 65)(37 91 41 95 45 87)(38 88 46 96 42 92)(39 93 43 85 47 89)(40 90 48 86 44 94)(49 75 57 83 53 79)(50 80 54 84 58 76)(51 77 59 73 55 81)(52 82 56 74 60 78)
(1 80)(2 73)(3 82)(4 75)(5 84)(6 77)(7 74)(8 79)(9 76)(10 81)(11 78)(12 83)(13 41)(14 46)(15 43)(16 48)(17 45)(18 38)(19 47)(20 40)(21 37)(22 42)(23 39)(24 44)(25 51)(26 60)(27 53)(28 50)(29 55)(30 52)(31 57)(32 54)(33 59)(34 56)(35 49)(36 58)(61 94)(62 91)(63 96)(64 93)(65 86)(66 95)(67 88)(68 85)(69 90)(70 87)(71 92)(72 89)
(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 40 7 46)(2 39 8 45)(3 38 9 44)(4 37 10 43)(5 48 11 42)(6 47 12 41)(13 51 19 57)(14 50 20 56)(15 49 21 55)(16 60 22 54)(17 59 23 53)(18 58 24 52)(25 85 31 91)(26 96 32 90)(27 95 33 89)(28 94 34 88)(29 93 35 87)(30 92 36 86)(61 82 67 76)(62 81 68 75)(63 80 69 74)(64 79 70 73)(65 78 71 84)(66 77 72 83)

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

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

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

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 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 C2 S3 S3 D4 Q8 D6 D6 D6 C4○D4 C3⋊D4 Dic6 C4○D12 S32 S3×D4 S3×Q8 C2×S32 S3×Dic6 D6.D6 S3×C3⋊D4 kernel D6⋊Dic6 D6⋊Dic3 Dic3⋊Dic3 C3×Dic3⋊C4 C6.Dic6 C2×C32⋊2Q8 S3×C2×C12 Dic3⋊C4 S3×C2×C4 C3×Dic3 S3×C6 C2×Dic3 C2×C12 C22×S3 C3×C6 Dic3 D6 C6 C2×C4 C6 C6 C22 C2 C2 C2 # reps 1 2 1 1 1 1 1 1 1 2 2 3 2 1 2 4 4 8 1 1 1 1 2 2 2

Matrix representation of D6⋊Dic6 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 12 1 0 0 0 0 12 0
,
 2 9 0 0 0 0 4 11 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 12 1
,
 7 3 0 0 0 0 10 10 0 0 0 0 0 0 10 3 0 0 0 0 10 7 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 11 11 0 0 0 0 9 2 0 0 0 0 0 0 2 4 0 0 0 0 2 11 0 0 0 0 0 0 0 1 0 0 0 0 1 0

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

D6⋊Dic6 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(288,499);
// by ID

G=gap.SmallGroup(288,499);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,120,219,142,1356,9414]);
// Polycyclic

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

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