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## G = D12⋊Dic3order 288 = 25·32

### 5th semidirect product of D12 and Dic3 acting via Dic3/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — D12⋊Dic3
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×S3×Dic3 — D12⋊Dic3
 Lower central C32 — C3×C6 — D12⋊Dic3
 Upper central C1 — C22 — C2×C4

Generators and relations for D12⋊Dic3
G = < a,b,c,d | a12=b2=c6=1, d2=c3, bab=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a6b, dcd-1=c-1 >

Subgroups: 682 in 203 conjugacy classes, 70 normal (32 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×8], S3 [×4], C6 [×6], C6 [×7], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], C32, Dic3 [×12], C12 [×4], C12 [×4], D6 [×4], D6 [×4], C2×C6 [×2], C2×C6 [×9], C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C3×S3 [×4], C3×C6 [×3], C4×S3 [×4], D12 [×4], C2×Dic3 [×2], C2×Dic3 [×10], C2×C12 [×2], C2×C12 [×3], C3×D4 [×4], C22×S3 [×2], C22×C6 [×2], C4×D4, C3×Dic3 [×2], C3⋊Dic3 [×2], C3⋊Dic3, C3×C12 [×2], S3×C6 [×4], S3×C6 [×4], C62, C4×Dic3 [×3], C4⋊Dic3, D6⋊C4 [×2], C6.D4 [×2], C3×C4⋊C4, S3×C2×C4 [×2], C2×D12, C22×Dic3 [×2], C6×D4, S3×Dic3 [×4], C3×D12 [×4], C6×Dic3 [×2], C2×C3⋊Dic3 [×2], C6×C12, S3×C2×C6 [×2], Dic35D4, D4×Dic3, D6⋊Dic3 [×2], C3×C4⋊Dic3, C4×C3⋊Dic3, C2×S3×Dic3 [×2], C6×D12, D12⋊Dic3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], D4 [×2], C23, Dic3 [×4], D6 [×6], C22×C4, C2×D4, C4○D4, C4×S3 [×2], C2×Dic3 [×6], C22×S3 [×2], C4×D4, S32, S3×C2×C4, S3×D4 [×2], D42S3, Q83S3, C22×Dic3, S3×Dic3 [×2], C2×S32, Dic35D4, D4×Dic3, D12⋊S3, D6⋊D6, C2×S3×Dic3, D12⋊Dic3

Smallest permutation representation of D12⋊Dic3
On 96 points
Generators in S96
(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 27)(2 26)(3 25)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 62)(14 61)(15 72)(16 71)(17 70)(18 69)(19 68)(20 67)(21 66)(22 65)(23 64)(24 63)(37 57)(38 56)(39 55)(40 54)(41 53)(42 52)(43 51)(44 50)(45 49)(46 60)(47 59)(48 58)(73 91)(74 90)(75 89)(76 88)(77 87)(78 86)(79 85)(80 96)(81 95)(82 94)(83 93)(84 92)
(1 17 9 13 5 21)(2 18 10 14 6 22)(3 19 11 15 7 23)(4 20 12 16 8 24)(25 68 29 72 33 64)(26 69 30 61 34 65)(27 70 31 62 35 66)(28 71 32 63 36 67)(37 94 41 86 45 90)(38 95 42 87 46 91)(39 96 43 88 47 92)(40 85 44 89 48 93)(49 74 57 82 53 78)(50 75 58 83 54 79)(51 76 59 84 55 80)(52 77 60 73 56 81)
(1 47 13 96)(2 42 14 91)(3 37 15 86)(4 44 16 93)(5 39 17 88)(6 46 18 95)(7 41 19 90)(8 48 20 85)(9 43 21 92)(10 38 22 87)(11 45 23 94)(12 40 24 89)(25 51 72 84)(26 58 61 79)(27 53 62 74)(28 60 63 81)(29 55 64 76)(30 50 65 83)(31 57 66 78)(32 52 67 73)(33 59 68 80)(34 54 69 75)(35 49 70 82)(36 56 71 77)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 12A ··· 12H 12I 12J 12K 12L order 1 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 6 ··· 6 6 6 6 6 6 6 6 12 ··· 12 12 12 12 12 size 1 1 1 1 6 6 6 6 2 2 4 2 2 6 6 6 6 9 9 9 9 18 18 2 ··· 2 4 4 4 12 12 12 12 4 ··· 4 12 12 12 12

48 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 4 type + + + + + + + + + - + + + + + - + - + image C1 C2 C2 C2 C2 C2 C4 S3 S3 D4 Dic3 D6 D6 D6 C4○D4 C4×S3 S32 S3×D4 D4⋊2S3 Q8⋊3S3 S3×Dic3 C2×S32 D12⋊S3 D6⋊D6 kernel D12⋊Dic3 D6⋊Dic3 C3×C4⋊Dic3 C4×C3⋊Dic3 C2×S3×Dic3 C6×D12 C3×D12 C4⋊Dic3 C2×D12 C3⋊Dic3 D12 C2×Dic3 C2×C12 C22×S3 C3×C6 C12 C2×C4 C6 C6 C6 C4 C22 C2 C2 # reps 1 2 1 1 2 1 8 1 1 2 4 2 2 2 2 4 1 2 1 1 2 1 2 2

Matrix representation of D12⋊Dic3 in GL6(𝔽13)

 0 12 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 2 6 0 0 0 0 6 11 0 0 0 0 0 0 0 12 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 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 12 12 0 0 0 0 1 0
,
 6 11 0 0 0 0 11 7 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 1 0 0 0 0 0 12 12

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

D12⋊Dic3 in GAP, Magma, Sage, TeX

D_{12}\rtimes {\rm Dic}_3
% in TeX

G:=Group("D12:Dic3");
// GroupNames label

G:=SmallGroup(288,546);
// by ID

G=gap.SmallGroup(288,546);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,422,219,100,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^6=1,d^2=c^3,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^6*b,d*c*d^-1=c^-1>;
// generators/relations

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