metabelian, supersoluble, monomial
Aliases: C12.16D12, C62.20D4, Dic6⋊3Dic3, (C3×C6).3Q16, C12.16(C4×S3), (C3×Dic6)⋊1C4, (C2×C12).78D6, (C3×C12).35D4, C4⋊Dic3.8S3, C4.7(S3×Dic3), C6.36(D6⋊C4), C6.5(D4.S3), (C2×Dic6).6S3, (C6×Dic6).2C2, C12.8(C2×Dic3), (C3×C6).10SD16, C6.4(C3⋊Q16), C3⋊3(C6.SD16), C3⋊2(Q8⋊2Dic3), C12.31(C3⋊D4), C2.7(D6⋊Dic3), (C6×C12).27C22, C6.5(Q8⋊2S3), C32⋊4(Q8⋊C4), C4.22(C3⋊D12), C6.6(C6.D4), C2.1(C32⋊2Q16), C2.2(Dic6⋊S3), C22.8(D6⋊S3), (C2×C4).100S32, (C3×C12).25(C2×C4), (C3×C4⋊Dic3).1C2, (C2×C6).49(C3⋊D4), (C2×C32⋊4C8).1C2, (C3×C6).32(C22⋊C4), SmallGroup(288,213)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6⋊Dic3
G = < a,b,c,d | a12=c6=1, b2=a6, d2=c3, bab-1=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a9b, dcd-1=c-1 >
Subgroups: 274 in 95 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C3, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, Q8, C32, Dic3, C12, C12, C2×C6, C2×C6, C4⋊C4, C2×C8, C2×Q8, C3×C6, C3⋊C8, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, Q8⋊C4, C3×Dic3, C3×C12, C62, C2×C3⋊C8, C4⋊Dic3, C3×C4⋊C4, C2×Dic6, C6×Q8, C32⋊4C8, C3×Dic6, C3×Dic6, C6×Dic3, C6×C12, C6.SD16, Q8⋊2Dic3, C3×C4⋊Dic3, C2×C32⋊4C8, C6×Dic6, Dic6⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, SD16, Q16, C4×S3, D12, C2×Dic3, C3⋊D4, Q8⋊C4, S32, D6⋊C4, D4.S3, Q8⋊2S3, C3⋊Q16, C6.D4, S3×Dic3, D6⋊S3, C3⋊D12, C6.SD16, Q8⋊2Dic3, Dic6⋊S3, C32⋊2Q16, D6⋊Dic3, Dic6⋊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 61 7 67)(2 72 8 66)(3 71 9 65)(4 70 10 64)(5 69 11 63)(6 68 12 62)(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 89 31 95)(26 88 32 94)(27 87 33 93)(28 86 34 92)(29 85 35 91)(30 96 36 90)(37 77 43 83)(38 76 44 82)(39 75 45 81)(40 74 46 80)(41 73 47 79)(42 84 48 78)
(1 23 5 15 9 19)(2 24 6 16 10 20)(3 13 7 17 11 21)(4 14 8 18 12 22)(25 41 33 37 29 45)(26 42 34 38 30 46)(27 43 35 39 31 47)(28 44 36 40 32 48)(49 65 57 61 53 69)(50 66 58 62 54 70)(51 67 59 63 55 71)(52 68 60 64 56 72)(73 93 77 85 81 89)(74 94 78 86 82 90)(75 95 79 87 83 91)(76 96 80 88 84 92)
(1 47 15 35)(2 42 16 30)(3 37 17 25)(4 44 18 32)(5 39 19 27)(6 46 20 34)(7 41 21 29)(8 48 22 36)(9 43 23 31)(10 38 24 26)(11 45 13 33)(12 40 14 28)(49 94 61 82)(50 89 62 77)(51 96 63 84)(52 91 64 79)(53 86 65 74)(54 93 66 81)(55 88 67 76)(56 95 68 83)(57 90 69 78)(58 85 70 73)(59 92 71 80)(60 87 72 75)
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), (1,61,7,67)(2,72,8,66)(3,71,9,65)(4,70,10,64)(5,69,11,63)(6,68,12,62)(13,51,19,57)(14,50,20,56)(15,49,21,55)(16,60,22,54)(17,59,23,53)(18,58,24,52)(25,89,31,95)(26,88,32,94)(27,87,33,93)(28,86,34,92)(29,85,35,91)(30,96,36,90)(37,77,43,83)(38,76,44,82)(39,75,45,81)(40,74,46,80)(41,73,47,79)(42,84,48,78), (1,23,5,15,9,19)(2,24,6,16,10,20)(3,13,7,17,11,21)(4,14,8,18,12,22)(25,41,33,37,29,45)(26,42,34,38,30,46)(27,43,35,39,31,47)(28,44,36,40,32,48)(49,65,57,61,53,69)(50,66,58,62,54,70)(51,67,59,63,55,71)(52,68,60,64,56,72)(73,93,77,85,81,89)(74,94,78,86,82,90)(75,95,79,87,83,91)(76,96,80,88,84,92), (1,47,15,35)(2,42,16,30)(3,37,17,25)(4,44,18,32)(5,39,19,27)(6,46,20,34)(7,41,21,29)(8,48,22,36)(9,43,23,31)(10,38,24,26)(11,45,13,33)(12,40,14,28)(49,94,61,82)(50,89,62,77)(51,96,63,84)(52,91,64,79)(53,86,65,74)(54,93,66,81)(55,88,67,76)(56,95,68,83)(57,90,69,78)(58,85,70,73)(59,92,71,80)(60,87,72,75)>;
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), (1,61,7,67)(2,72,8,66)(3,71,9,65)(4,70,10,64)(5,69,11,63)(6,68,12,62)(13,51,19,57)(14,50,20,56)(15,49,21,55)(16,60,22,54)(17,59,23,53)(18,58,24,52)(25,89,31,95)(26,88,32,94)(27,87,33,93)(28,86,34,92)(29,85,35,91)(30,96,36,90)(37,77,43,83)(38,76,44,82)(39,75,45,81)(40,74,46,80)(41,73,47,79)(42,84,48,78), (1,23,5,15,9,19)(2,24,6,16,10,20)(3,13,7,17,11,21)(4,14,8,18,12,22)(25,41,33,37,29,45)(26,42,34,38,30,46)(27,43,35,39,31,47)(28,44,36,40,32,48)(49,65,57,61,53,69)(50,66,58,62,54,70)(51,67,59,63,55,71)(52,68,60,64,56,72)(73,93,77,85,81,89)(74,94,78,86,82,90)(75,95,79,87,83,91)(76,96,80,88,84,92), (1,47,15,35)(2,42,16,30)(3,37,17,25)(4,44,18,32)(5,39,19,27)(6,46,20,34)(7,41,21,29)(8,48,22,36)(9,43,23,31)(10,38,24,26)(11,45,13,33)(12,40,14,28)(49,94,61,82)(50,89,62,77)(51,96,63,84)(52,91,64,79)(53,86,65,74)(54,93,66,81)(55,88,67,76)(56,95,68,83)(57,90,69,78)(58,85,70,73)(59,92,71,80)(60,87,72,75) );
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)], [(1,61,7,67),(2,72,8,66),(3,71,9,65),(4,70,10,64),(5,69,11,63),(6,68,12,62),(13,51,19,57),(14,50,20,56),(15,49,21,55),(16,60,22,54),(17,59,23,53),(18,58,24,52),(25,89,31,95),(26,88,32,94),(27,87,33,93),(28,86,34,92),(29,85,35,91),(30,96,36,90),(37,77,43,83),(38,76,44,82),(39,75,45,81),(40,74,46,80),(41,73,47,79),(42,84,48,78)], [(1,23,5,15,9,19),(2,24,6,16,10,20),(3,13,7,17,11,21),(4,14,8,18,12,22),(25,41,33,37,29,45),(26,42,34,38,30,46),(27,43,35,39,31,47),(28,44,36,40,32,48),(49,65,57,61,53,69),(50,66,58,62,54,70),(51,67,59,63,55,71),(52,68,60,64,56,72),(73,93,77,85,81,89),(74,94,78,86,82,90),(75,95,79,87,83,91),(76,96,80,88,84,92)], [(1,47,15,35),(2,42,16,30),(3,37,17,25),(4,44,18,32),(5,39,19,27),(6,46,20,34),(7,41,21,29),(8,48,22,36),(9,43,23,31),(10,38,24,26),(11,45,13,33),(12,40,14,28),(49,94,61,82),(50,89,62,77),(51,96,63,84),(52,91,64,79),(53,86,65,74),(54,93,66,81),(55,88,67,76),(56,95,68,83),(57,90,69,78),(58,85,70,73),(59,92,71,80),(60,87,72,75)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | 6H | 6I | 8A | 8B | 8C | 8D | 12A | ··· | 12H | 12I | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 4 | 4 | 4 | 18 | 18 | 18 | 18 | 4 | ··· | 4 | 12 | ··· | 12 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | - | + | + | - | + | - | - | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | S3 | D4 | D4 | Dic3 | D6 | SD16 | Q16 | C4×S3 | D12 | C3⋊D4 | C3⋊D4 | S32 | D4.S3 | Q8⋊2S3 | C3⋊Q16 | S3×Dic3 | C3⋊D12 | D6⋊S3 | Dic6⋊S3 | C32⋊2Q16 |
| kernel | Dic6⋊Dic3 | C3×C4⋊Dic3 | C2×C32⋊4C8 | C6×Dic6 | C3×Dic6 | C4⋊Dic3 | C2×Dic6 | C3×C12 | C62 | Dic6 | C2×C12 | C3×C6 | C3×C6 | C12 | C12 | C12 | C2×C6 | C2×C4 | C6 | C6 | C6 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of Dic6⋊Dic3 ►in GL6(𝔽73)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 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 | 72 | 0 |
| 6 | 67 | 0 | 0 | 0 | 0 |
| 67 | 67 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 46 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 43 | 13 |
| 0 | 0 | 0 | 0 | 60 | 30 |
G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,0,0,1,0],[6,67,0,0,0,0,67,67,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,72,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,46,0,0,0,0,46,0,0,0,0,0,0,0,43,60,0,0,0,0,13,30] >;
Dic6⋊Dic3 in GAP, Magma, Sage, TeX
{\rm Dic}_6\rtimes {\rm Dic}_3 % in TeX
G:=Group("Dic6:Dic3"); // GroupNames label
G:=SmallGroup(288,213);
// by ID
G=gap.SmallGroup(288,213);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,141,36,675,346,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^6=1,b^2=a^6,d^2=c^3,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^9*b,d*c*d^-1=c^-1>;
// generators/relations