metabelian, supersoluble, monomial
Aliases: D12.12D6, C3⋊C8.17D6, Q8.11S32, Q8⋊2S3⋊7S3, Q8⋊3S3⋊5S3, (S3×C6).14D4, (C4×S3).24D6, C6.159(S3×D4), C32⋊2D8⋊9C2, (C3×Q8).43D6, D12⋊5S3⋊6C2, D6.3(C3⋊D4), C32⋊13(C4○D8), C32⋊7Q16⋊4C2, C3⋊4(Q8.13D6), C3⋊7(Q8.7D6), (C3×C12).24C23, C12.24(C22×S3), (C3×Dic3).41D4, D12.S3⋊14C2, (S3×C12).23C22, (C3×D12).21C22, Dic3.22(C3⋊D4), (Q8×C32).6C22, C32⋊4C8.12C22, C32⋊4Q8.14C22, (S3×C3⋊C8)⋊7C2, C4.24(C2×S32), C6.55(C2×C3⋊D4), C2.33(S3×C3⋊D4), (C3×Q8⋊3S3)⋊2C2, (C3×Q8⋊2S3)⋊7C2, (C3×C6).139(C2×D4), (C3×C3⋊C8).19C22, SmallGroup(288,595)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.12D6
G = < a,b,c,d | a12=b2=c6=1, d2=a6, bab=dad-1=a-1, cac-1=a5, cbc-1=a10b, dbd-1=ab, dcd-1=a6c-1 >
Subgroups: 498 in 133 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, D4, Q8, Q8, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C8, D8, SD16, Q16, C4○D4, C3×S3, C3×C6, C3⋊C8, C3⋊C8, C24, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C3×Q8, C4○D8, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, S3×C6, S3×C8, C24⋊C2, C2×C3⋊C8, D4⋊S3, D4.S3, Q8⋊2S3, C3⋊Q16, C3×SD16, C4○D12, D4⋊2S3, Q8⋊3S3, C3×C4○D4, C3×C3⋊C8, C32⋊4C8, S3×Dic3, D6⋊S3, S3×C12, S3×C12, C3×D12, C3×D12, C32⋊4Q8, Q8×C32, Q8.7D6, Q8.13D6, S3×C3⋊C8, C32⋊2D8, D12.S3, C3×Q8⋊2S3, C32⋊7Q16, D12⋊5S3, C3×Q8⋊3S3, D12.12D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C4○D8, S32, S3×D4, C2×C3⋊D4, C2×S32, Q8.7D6, Q8.13D6, S3×C3⋊D4, D12.12D6
►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 67)(2 66)(3 65)(4 64)(5 63)(6 62)(7 61)(8 72)(9 71)(10 70)(11 69)(12 68)(13 55)(14 54)(15 53)(16 52)(17 51)(18 50)(19 49)(20 60)(21 59)(22 58)(23 57)(24 56)(25 85)(26 96)(27 95)(28 94)(29 93)(30 92)(31 91)(32 90)(33 89)(34 88)(35 87)(36 86)(37 84)(38 83)(39 82)(40 81)(41 80)(42 79)(43 78)(44 77)(45 76)(46 75)(47 74)(48 73)
(1 60 5 56 9 52)(2 53 6 49 10 57)(3 58 7 54 11 50)(4 51 8 59 12 55)(13 66 17 62 21 70)(14 71 18 67 22 63)(15 64 19 72 23 68)(16 69 20 65 24 61)(25 75 33 79 29 83)(26 80 34 84 30 76)(27 73 35 77 31 81)(28 78 36 82 32 74)(37 94 45 86 41 90)(38 87 46 91 42 95)(39 92 47 96 43 88)(40 85 48 89 44 93)
(1 74 7 80)(2 73 8 79)(3 84 9 78)(4 83 10 77)(5 82 11 76)(6 81 12 75)(13 90 19 96)(14 89 20 95)(15 88 21 94)(16 87 22 93)(17 86 23 92)(18 85 24 91)(25 55 31 49)(26 54 32 60)(27 53 33 59)(28 52 34 58)(29 51 35 57)(30 50 36 56)(37 70 43 64)(38 69 44 63)(39 68 45 62)(40 67 46 61)(41 66 47 72)(42 65 48 71)
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), (1,67)(2,66)(3,65)(4,64)(5,63)(6,62)(7,61)(8,72)(9,71)(10,70)(11,69)(12,68)(13,55)(14,54)(15,53)(16,52)(17,51)(18,50)(19,49)(20,60)(21,59)(22,58)(23,57)(24,56)(25,85)(26,96)(27,95)(28,94)(29,93)(30,92)(31,91)(32,90)(33,89)(34,88)(35,87)(36,86)(37,84)(38,83)(39,82)(40,81)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73), (1,60,5,56,9,52)(2,53,6,49,10,57)(3,58,7,54,11,50)(4,51,8,59,12,55)(13,66,17,62,21,70)(14,71,18,67,22,63)(15,64,19,72,23,68)(16,69,20,65,24,61)(25,75,33,79,29,83)(26,80,34,84,30,76)(27,73,35,77,31,81)(28,78,36,82,32,74)(37,94,45,86,41,90)(38,87,46,91,42,95)(39,92,47,96,43,88)(40,85,48,89,44,93), (1,74,7,80)(2,73,8,79)(3,84,9,78)(4,83,10,77)(5,82,11,76)(6,81,12,75)(13,90,19,96)(14,89,20,95)(15,88,21,94)(16,87,22,93)(17,86,23,92)(18,85,24,91)(25,55,31,49)(26,54,32,60)(27,53,33,59)(28,52,34,58)(29,51,35,57)(30,50,36,56)(37,70,43,64)(38,69,44,63)(39,68,45,62)(40,67,46,61)(41,66,47,72)(42,65,48,71)>;
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), (1,67)(2,66)(3,65)(4,64)(5,63)(6,62)(7,61)(8,72)(9,71)(10,70)(11,69)(12,68)(13,55)(14,54)(15,53)(16,52)(17,51)(18,50)(19,49)(20,60)(21,59)(22,58)(23,57)(24,56)(25,85)(26,96)(27,95)(28,94)(29,93)(30,92)(31,91)(32,90)(33,89)(34,88)(35,87)(36,86)(37,84)(38,83)(39,82)(40,81)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73), (1,60,5,56,9,52)(2,53,6,49,10,57)(3,58,7,54,11,50)(4,51,8,59,12,55)(13,66,17,62,21,70)(14,71,18,67,22,63)(15,64,19,72,23,68)(16,69,20,65,24,61)(25,75,33,79,29,83)(26,80,34,84,30,76)(27,73,35,77,31,81)(28,78,36,82,32,74)(37,94,45,86,41,90)(38,87,46,91,42,95)(39,92,47,96,43,88)(40,85,48,89,44,93), (1,74,7,80)(2,73,8,79)(3,84,9,78)(4,83,10,77)(5,82,11,76)(6,81,12,75)(13,90,19,96)(14,89,20,95)(15,88,21,94)(16,87,22,93)(17,86,23,92)(18,85,24,91)(25,55,31,49)(26,54,32,60)(27,53,33,59)(28,52,34,58)(29,51,35,57)(30,50,36,56)(37,70,43,64)(38,69,44,63)(39,68,45,62)(40,67,46,61)(41,66,47,72)(42,65,48,71) );
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)], [(1,67),(2,66),(3,65),(4,64),(5,63),(6,62),(7,61),(8,72),(9,71),(10,70),(11,69),(12,68),(13,55),(14,54),(15,53),(16,52),(17,51),(18,50),(19,49),(20,60),(21,59),(22,58),(23,57),(24,56),(25,85),(26,96),(27,95),(28,94),(29,93),(30,92),(31,91),(32,90),(33,89),(34,88),(35,87),(36,86),(37,84),(38,83),(39,82),(40,81),(41,80),(42,79),(43,78),(44,77),(45,76),(46,75),(47,74),(48,73)], [(1,60,5,56,9,52),(2,53,6,49,10,57),(3,58,7,54,11,50),(4,51,8,59,12,55),(13,66,17,62,21,70),(14,71,18,67,22,63),(15,64,19,72,23,68),(16,69,20,65,24,61),(25,75,33,79,29,83),(26,80,34,84,30,76),(27,73,35,77,31,81),(28,78,36,82,32,74),(37,94,45,86,41,90),(38,87,46,91,42,95),(39,92,47,96,43,88),(40,85,48,89,44,93)], [(1,74,7,80),(2,73,8,79),(3,84,9,78),(4,83,10,77),(5,82,11,76),(6,81,12,75),(13,90,19,96),(14,89,20,95),(15,88,21,94),(16,87,22,93),(17,86,23,92),(18,85,24,91),(25,55,31,49),(26,54,32,60),(27,53,33,59),(28,52,34,58),(29,51,35,57),(30,50,36,56),(37,70,43,64),(38,69,44,63),(39,68,45,62),(40,67,46,61),(41,66,47,72),(42,65,48,71)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 24A | 24B |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 |
| size | 1 | 1 | 6 | 12 | 12 | 2 | 2 | 4 | 2 | 3 | 3 | 4 | 36 | 2 | 2 | 4 | 12 | 12 | 12 | 24 | 6 | 6 | 18 | 18 | 4 | 4 | 4 | 4 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D4 | D4 | D6 | D6 | D6 | D6 | C3⋊D4 | C3⋊D4 | C4○D8 | S32 | S3×D4 | C2×S32 | Q8.7D6 | Q8.13D6 | S3×C3⋊D4 | D12.12D6 |
| kernel | D12.12D6 | S3×C3⋊C8 | C32⋊2D8 | D12.S3 | C3×Q8⋊2S3 | C32⋊7Q16 | D12⋊5S3 | C3×Q8⋊3S3 | Q8⋊2S3 | Q8⋊3S3 | C3×Dic3 | S3×C6 | C3⋊C8 | C4×S3 | D12 | C3×Q8 | Dic3 | D6 | C32 | Q8 | C6 | C4 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 1 | 1 | 2 | 2 | 2 | 1 |
Matrix representation of D12.12D6 ►in GL6(𝔽73)
| 1 | 48 | 0 | 0 | 0 | 0 |
| 3 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 52 | 16 | 0 | 0 | 0 | 0 |
| 9 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 27 | 55 | 0 | 0 | 0 | 0 |
| 8 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 37 | 62 | 0 | 0 | 0 | 0 |
| 25 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
G:=sub<GL(6,GF(73))| [1,3,0,0,0,0,48,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[52,9,0,0,0,0,16,21,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[27,8,0,0,0,0,55,46,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,72,0],[37,25,0,0,0,0,62,36,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,0,1] >;
D12.12D6 in GAP, Magma, Sage, TeX
D_{12}._{12}D_6 % in TeX
G:=Group("D12.12D6"); // GroupNames label
G:=SmallGroup(288,595);
// by ID
G=gap.SmallGroup(288,595);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,422,135,100,346,185,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^6=1,d^2=a^6,b*a*b=d*a*d^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^10*b,d*b*d^-1=a*b,d*c*d^-1=a^6*c^-1>;
// generators/relations