metabelian, supersoluble, monomial
Aliases: D12⋊5Dic3, C62.68C23, C12⋊2(C4×S3), C32⋊7(C4×D4), (C3×D12)⋊8C4, C3⋊Dic3⋊7D4, C3⋊2(D4×Dic3), C4⋊2(S3×Dic3), C6.37(S3×D4), C4⋊Dic3⋊13S3, D6⋊2(C2×Dic3), (C2×D12).9S3, C12⋊2(C2×Dic3), D6⋊Dic3⋊21C2, (C6×D12).11C2, (C2×C12).137D6, C3⋊5(Dic3⋊5D4), C2.1(D6⋊D6), (C2×Dic3).72D6, (C22×S3).41D6, C2.5(D12⋊S3), C6.10(D4⋊2S3), (C6×C12).103C22, C6.35(Q8⋊3S3), C6.13(C22×Dic3), (C6×Dic3).65C22, C6.92(S3×C2×C4), (S3×C6)⋊5(C2×C4), (C3×C12)⋊7(C2×C4), (C2×C4).117S32, (C4×C3⋊Dic3)⋊5C2, C22.37(C2×S32), (C2×S3×Dic3)⋊15C2, (C3×C6).53(C2×D4), C2.14(C2×S3×Dic3), (C3×C4⋊Dic3)⋊15C2, (S3×C2×C6).24C22, (C3×C6).41(C4○D4), (C3×C6).58(C22×C4), (C2×C6).87(C22×S3), (C2×C3⋊Dic3).131C22, SmallGroup(288,546)
Series: Derived ►Chief ►Lower central ►Upper central
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, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C32, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×S3, C3×C6, C4×S3, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C4×D4, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C62, C4×Dic3, C4⋊Dic3, D6⋊C4, C6.D4, C3×C4⋊C4, S3×C2×C4, C2×D12, C22×Dic3, C6×D4, S3×Dic3, C3×D12, C6×Dic3, C2×C3⋊Dic3, C6×C12, S3×C2×C6, Dic3⋊5D4, D4×Dic3, D6⋊Dic3, C3×C4⋊Dic3, C4×C3⋊Dic3, C2×S3×Dic3, C6×D12, D12⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C22×C4, C2×D4, C4○D4, C4×S3, C2×Dic3, C22×S3, C4×D4, S32, S3×C2×C4, S3×D4, D4⋊2S3, Q8⋊3S3, C22×Dic3, S3×Dic3, C2×S32, Dic3⋊5D4, D4×Dic3, D12⋊S3, D6⋊D6, C2×S3×Dic3, D12⋊Dic3
►On 96 points
(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)(2 81)(3 80)(4 79)(5 78)(6 77)(7 76)(8 75)(9 74)(10 73)(11 84)(12 83)(13 28)(14 27)(15 26)(16 25)(17 36)(18 35)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(37 66)(38 65)(39 64)(40 63)(41 62)(42 61)(43 72)(44 71)(45 70)(46 69)(47 68)(48 67)(49 92)(50 91)(51 90)(52 89)(53 88)(54 87)(55 86)(56 85)(57 96)(58 95)(59 94)(60 93)
(1 35 9 31 5 27)(2 36 10 32 6 28)(3 25 11 33 7 29)(4 26 12 34 8 30)(13 81 17 73 21 77)(14 82 18 74 22 78)(15 83 19 75 23 79)(16 84 20 76 24 80)(37 91 41 95 45 87)(38 92 42 96 46 88)(39 93 43 85 47 89)(40 94 44 86 48 90)(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 94 31 48)(2 89 32 43)(3 96 33 38)(4 91 34 45)(5 86 35 40)(6 93 36 47)(7 88 25 42)(8 95 26 37)(9 90 27 44)(10 85 28 39)(11 92 29 46)(12 87 30 41)(13 70 73 50)(14 65 74 57)(15 72 75 52)(16 67 76 59)(17 62 77 54)(18 69 78 49)(19 64 79 56)(20 71 80 51)(21 66 81 58)(22 61 82 53)(23 68 83 60)(24 63 84 55)
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), (1,82)(2,81)(3,80)(4,79)(5,78)(6,77)(7,76)(8,75)(9,74)(10,73)(11,84)(12,83)(13,28)(14,27)(15,26)(16,25)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(37,66)(38,65)(39,64)(40,63)(41,62)(42,61)(43,72)(44,71)(45,70)(46,69)(47,68)(48,67)(49,92)(50,91)(51,90)(52,89)(53,88)(54,87)(55,86)(56,85)(57,96)(58,95)(59,94)(60,93), (1,35,9,31,5,27)(2,36,10,32,6,28)(3,25,11,33,7,29)(4,26,12,34,8,30)(13,81,17,73,21,77)(14,82,18,74,22,78)(15,83,19,75,23,79)(16,84,20,76,24,80)(37,91,41,95,45,87)(38,92,42,96,46,88)(39,93,43,85,47,89)(40,94,44,86,48,90)(49,61,57,69,53,65)(50,62,58,70,54,66)(51,63,59,71,55,67)(52,64,60,72,56,68), (1,94,31,48)(2,89,32,43)(3,96,33,38)(4,91,34,45)(5,86,35,40)(6,93,36,47)(7,88,25,42)(8,95,26,37)(9,90,27,44)(10,85,28,39)(11,92,29,46)(12,87,30,41)(13,70,73,50)(14,65,74,57)(15,72,75,52)(16,67,76,59)(17,62,77,54)(18,69,78,49)(19,64,79,56)(20,71,80,51)(21,66,81,58)(22,61,82,53)(23,68,83,60)(24,63,84,55)>;
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), (1,82)(2,81)(3,80)(4,79)(5,78)(6,77)(7,76)(8,75)(9,74)(10,73)(11,84)(12,83)(13,28)(14,27)(15,26)(16,25)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(37,66)(38,65)(39,64)(40,63)(41,62)(42,61)(43,72)(44,71)(45,70)(46,69)(47,68)(48,67)(49,92)(50,91)(51,90)(52,89)(53,88)(54,87)(55,86)(56,85)(57,96)(58,95)(59,94)(60,93), (1,35,9,31,5,27)(2,36,10,32,6,28)(3,25,11,33,7,29)(4,26,12,34,8,30)(13,81,17,73,21,77)(14,82,18,74,22,78)(15,83,19,75,23,79)(16,84,20,76,24,80)(37,91,41,95,45,87)(38,92,42,96,46,88)(39,93,43,85,47,89)(40,94,44,86,48,90)(49,61,57,69,53,65)(50,62,58,70,54,66)(51,63,59,71,55,67)(52,64,60,72,56,68), (1,94,31,48)(2,89,32,43)(3,96,33,38)(4,91,34,45)(5,86,35,40)(6,93,36,47)(7,88,25,42)(8,95,26,37)(9,90,27,44)(10,85,28,39)(11,92,29,46)(12,87,30,41)(13,70,73,50)(14,65,74,57)(15,72,75,52)(16,67,76,59)(17,62,77,54)(18,69,78,49)(19,64,79,56)(20,71,80,51)(21,66,81,58)(22,61,82,53)(23,68,83,60)(24,63,84,55) );
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)], [(1,82),(2,81),(3,80),(4,79),(5,78),(6,77),(7,76),(8,75),(9,74),(10,73),(11,84),(12,83),(13,28),(14,27),(15,26),(16,25),(17,36),(18,35),(19,34),(20,33),(21,32),(22,31),(23,30),(24,29),(37,66),(38,65),(39,64),(40,63),(41,62),(42,61),(43,72),(44,71),(45,70),(46,69),(47,68),(48,67),(49,92),(50,91),(51,90),(52,89),(53,88),(54,87),(55,86),(56,85),(57,96),(58,95),(59,94),(60,93)], [(1,35,9,31,5,27),(2,36,10,32,6,28),(3,25,11,33,7,29),(4,26,12,34,8,30),(13,81,17,73,21,77),(14,82,18,74,22,78),(15,83,19,75,23,79),(16,84,20,76,24,80),(37,91,41,95,45,87),(38,92,42,96,46,88),(39,93,43,85,47,89),(40,94,44,86,48,90),(49,61,57,69,53,65),(50,62,58,70,54,66),(51,63,59,71,55,67),(52,64,60,72,56,68)], [(1,94,31,48),(2,89,32,43),(3,96,33,38),(4,91,34,45),(5,86,35,40),(6,93,36,47),(7,88,25,42),(8,95,26,37),(9,90,27,44),(10,85,28,39),(11,92,29,46),(12,87,30,41),(13,70,73,50),(14,65,74,57),(15,72,75,52),(16,67,76,59),(17,62,77,54),(18,69,78,49),(19,64,79,56),(20,71,80,51),(21,66,81,58),(22,61,82,53),(23,68,83,60),(24,63,84,55)]])
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