metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.3Dic6, C42.46D6, (C4×D4).4S3, C3⋊6(D4.Q8), (C3×D4).3Q8, C4⋊C4.240D6, C12⋊C8⋊19C2, (D4×C12).4C2, (C2×C12).59D4, C12.27(C2×Q8), (C2×D4).187D6, C6.87(C4○D8), C6.Q16⋊31C2, C4.11(C2×Dic6), C12.47(C4○D4), C4.61(C4○D12), C6.86(C8⋊C22), (C4×C12).80C22, C12.6Q8⋊11C2, C12.Q8⋊31C2, D4⋊Dic3.9C2, C6.63(C22⋊Q8), (C2×C12).334C23, C2.8(D12⋊6C22), (C6×D4).229C22, C2.10(Q8.13D6), C4⋊Dic3.138C22, C2.14(C12.48D4), (C2×C6).465(C2×D4), (C2×C3⋊C8).91C22, (C2×C4).216(C3⋊D4), (C3×C4⋊C4).271C22, (C2×C4).434(C22×S3), C22.148(C2×C3⋊D4), SmallGroup(192,568)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.3Dic6
G = < a,b,c,d | a4=b2=c12=1, d2=a2c6, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2c-1 >
Subgroups: 248 in 102 conjugacy classes, 43 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C22×C6, D4⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×D4, C42.C2, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, D4.Q8, C12⋊C8, C6.Q16, C12.Q8, D4⋊Dic3, C12.6Q8, D4×C12, D4.3Dic6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, Dic6, C3⋊D4, C22×S3, C22⋊Q8, C4○D8, C8⋊C22, C2×Dic6, C4○D12, C2×C3⋊D4, D4.Q8, C12.48D4, D12⋊6C22, Q8.13D6, D4.3Dic6
►On 96 points
Generators in S96
(1 78 72 28)(2 79 61 29)(3 80 62 30)(4 81 63 31)(5 82 64 32)(6 83 65 33)(7 84 66 34)(8 73 67 35)(9 74 68 36)(10 75 69 25)(11 76 70 26)(12 77 71 27)(13 47 85 49)(14 48 86 50)(15 37 87 51)(16 38 88 52)(17 39 89 53)(18 40 90 54)(19 41 91 55)(20 42 92 56)(21 43 93 57)(22 44 94 58)(23 45 95 59)(24 46 96 60)
(1 84)(2 73)(3 74)(4 75)(5 76)(6 77)(7 78)(8 79)(9 80)(10 81)(11 82)(12 83)(13 91)(14 92)(15 93)(16 94)(17 95)(18 96)(19 85)(20 86)(21 87)(22 88)(23 89)(24 90)(25 63)(26 64)(27 65)(28 66)(29 67)(30 68)(31 69)(32 70)(33 71)(34 72)(35 61)(36 62)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)
(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 51 66 43)(2 48 67 56)(3 49 68 41)(4 46 69 54)(5 59 70 39)(6 44 71 52)(7 57 72 37)(8 42 61 50)(9 55 62 47)(10 40 63 60)(11 53 64 45)(12 38 65 58)(13 74 91 30)(14 35 92 79)(15 84 93 28)(16 33 94 77)(17 82 95 26)(18 31 96 75)(19 80 85 36)(20 29 86 73)(21 78 87 34)(22 27 88 83)(23 76 89 32)(24 25 90 81)
G:=sub<Sym(96)| (1,78,72,28)(2,79,61,29)(3,80,62,30)(4,81,63,31)(5,82,64,32)(6,83,65,33)(7,84,66,34)(8,73,67,35)(9,74,68,36)(10,75,69,25)(11,76,70,26)(12,77,71,27)(13,47,85,49)(14,48,86,50)(15,37,87,51)(16,38,88,52)(17,39,89,53)(18,40,90,54)(19,41,91,55)(20,42,92,56)(21,43,93,57)(22,44,94,58)(23,45,95,59)(24,46,96,60), (1,84)(2,73)(3,74)(4,75)(5,76)(6,77)(7,78)(8,79)(9,80)(10,81)(11,82)(12,83)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,85)(20,86)(21,87)(22,88)(23,89)(24,90)(25,63)(26,64)(27,65)(28,66)(29,67)(30,68)(31,69)(32,70)(33,71)(34,72)(35,61)(36,62)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60), (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), (1,51,66,43)(2,48,67,56)(3,49,68,41)(4,46,69,54)(5,59,70,39)(6,44,71,52)(7,57,72,37)(8,42,61,50)(9,55,62,47)(10,40,63,60)(11,53,64,45)(12,38,65,58)(13,74,91,30)(14,35,92,79)(15,84,93,28)(16,33,94,77)(17,82,95,26)(18,31,96,75)(19,80,85,36)(20,29,86,73)(21,78,87,34)(22,27,88,83)(23,76,89,32)(24,25,90,81)>;
G:=Group( (1,78,72,28)(2,79,61,29)(3,80,62,30)(4,81,63,31)(5,82,64,32)(6,83,65,33)(7,84,66,34)(8,73,67,35)(9,74,68,36)(10,75,69,25)(11,76,70,26)(12,77,71,27)(13,47,85,49)(14,48,86,50)(15,37,87,51)(16,38,88,52)(17,39,89,53)(18,40,90,54)(19,41,91,55)(20,42,92,56)(21,43,93,57)(22,44,94,58)(23,45,95,59)(24,46,96,60), (1,84)(2,73)(3,74)(4,75)(5,76)(6,77)(7,78)(8,79)(9,80)(10,81)(11,82)(12,83)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,85)(20,86)(21,87)(22,88)(23,89)(24,90)(25,63)(26,64)(27,65)(28,66)(29,67)(30,68)(31,69)(32,70)(33,71)(34,72)(35,61)(36,62)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60), (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), (1,51,66,43)(2,48,67,56)(3,49,68,41)(4,46,69,54)(5,59,70,39)(6,44,71,52)(7,57,72,37)(8,42,61,50)(9,55,62,47)(10,40,63,60)(11,53,64,45)(12,38,65,58)(13,74,91,30)(14,35,92,79)(15,84,93,28)(16,33,94,77)(17,82,95,26)(18,31,96,75)(19,80,85,36)(20,29,86,73)(21,78,87,34)(22,27,88,83)(23,76,89,32)(24,25,90,81) );
G=PermutationGroup([[(1,78,72,28),(2,79,61,29),(3,80,62,30),(4,81,63,31),(5,82,64,32),(6,83,65,33),(7,84,66,34),(8,73,67,35),(9,74,68,36),(10,75,69,25),(11,76,70,26),(12,77,71,27),(13,47,85,49),(14,48,86,50),(15,37,87,51),(16,38,88,52),(17,39,89,53),(18,40,90,54),(19,41,91,55),(20,42,92,56),(21,43,93,57),(22,44,94,58),(23,45,95,59),(24,46,96,60)], [(1,84),(2,73),(3,74),(4,75),(5,76),(6,77),(7,78),(8,79),(9,80),(10,81),(11,82),(12,83),(13,91),(14,92),(15,93),(16,94),(17,95),(18,96),(19,85),(20,86),(21,87),(22,88),(23,89),(24,90),(25,63),(26,64),(27,65),(28,66),(29,67),(30,68),(31,69),(32,70),(33,71),(34,72),(35,61),(36,62),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48),(49,55),(50,56),(51,57),(52,58),(53,59),(54,60)], [(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)], [(1,51,66,43),(2,48,67,56),(3,49,68,41),(4,46,69,54),(5,59,70,39),(6,44,71,52),(7,57,72,37),(8,42,61,50),(9,55,62,47),(10,40,63,60),(11,53,64,45),(12,38,65,58),(13,74,91,30),(14,35,92,79),(15,84,93,28),(16,33,94,77),(17,82,95,26),(18,31,96,75),(19,80,85,36),(20,29,86,73),(21,78,87,34),(22,27,88,83),(23,76,89,32),(24,25,90,81)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 24 | 24 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | - | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D6 | D6 | D6 | C4○D4 | C3⋊D4 | Dic6 | C4○D8 | C4○D12 | C8⋊C22 | D12⋊6C22 | Q8.13D6 |
| kernel | D4.3Dic6 | C12⋊C8 | C6.Q16 | C12.Q8 | D4⋊Dic3 | C12.6Q8 | D4×C12 | C4×D4 | C2×C12 | C3×D4 | C42 | C4⋊C4 | C2×D4 | C12 | C2×C4 | D4 | C6 | C4 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 1 | 2 | 2 |
Matrix representation of D4.3Dic6 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 71 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 40 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 71 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 25 | 65 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 27 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 0 | 0 | 46 |
| 40 | 2 | 0 | 0 | 0 | 0 |
| 40 | 33 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 71 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 0 | 0 | 0 | 0 | 16 | 0 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,71,72],[72,40,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,71,72],[9,25,0,0,0,0,0,65,0,0,0,0,0,0,46,27,0,0,0,0,0,27,0,0,0,0,0,0,46,0,0,0,0,0,0,46],[40,40,0,0,0,0,2,33,0,0,0,0,0,0,72,1,0,0,0,0,71,1,0,0,0,0,0,0,0,16,0,0,0,0,32,0] >;
D4.3Dic6 in GAP, Magma, Sage, TeX
D_4._3{\rm Dic}_6 % in TeX
G:=Group("D4.3Dic6"); // GroupNames label
G:=SmallGroup(192,568);
// by ID
G=gap.SmallGroup(192,568);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,344,254,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^12=1,d^2=a^2*c^6,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=a^2*c^-1>;
// generators/relations