metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6.37D4, (C2×C6)⋊2Q16, C4⋊C4.69D6, C4.103(S3×D4), (C2×C12).79D4, (C2×Q8).55D6, C6.38(C2×Q16), C22⋊Q8.5S3, C6.49C22≀C2, C12.156(C2×D4), C3⋊3(C22⋊Q16), (C22×C6).96D4, C6.SD16⋊38C2, (C22×C4).146D6, C22⋊3(C3⋊Q16), (C6×Q8).49C22, C2.17(C23⋊2D6), (C2×C12).369C23, C12.55D4.9C2, C23.70(C3⋊D4), C2.15(Q8.14D6), C6.117(C8.C22), (C22×Dic6).13C2, (C22×C12).173C22, (C2×Dic6).271C22, (C2×C3⋊Q16)⋊9C2, C2.9(C2×C3⋊Q16), (C2×C6).500(C2×D4), (C3×C22⋊Q8).4C2, (C2×C4).57(C3⋊D4), (C2×C3⋊C8).117C22, (C3×C4⋊C4).116C22, (C2×C4).469(C22×S3), C22.175(C2×C3⋊D4), SmallGroup(192,609)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6.37D4
G = < a,b,c,d | a12=c4=d2=1, b2=a6, bab-1=a-1, cac-1=a7, ad=da, cbc-1=a9b, bd=db, dcd=a6c-1 >
Subgroups: 384 in 148 conjugacy classes, 47 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, Q16, C22×C4, C22×C4, C2×Q8, C2×Q8, C3⋊C8, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×C6, C22⋊C8, Q8⋊C4, C22⋊Q8, C2×Q16, C22×Q8, C2×C3⋊C8, C3⋊Q16, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C2×Dic6, C22×Dic3, C22×C12, C6×Q8, C22⋊Q16, C6.SD16, C12.55D4, C2×C3⋊Q16, C3×C22⋊Q8, C22×Dic6, Dic6.37D4
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, C3⋊D4, C22×S3, C22≀C2, C2×Q16, C8.C22, C3⋊Q16, S3×D4, C2×C3⋊D4, C22⋊Q16, C23⋊2D6, C2×C3⋊Q16, Q8.14D6, Dic6.37D4
►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 75 7 81)(2 74 8 80)(3 73 9 79)(4 84 10 78)(5 83 11 77)(6 82 12 76)(13 34 19 28)(14 33 20 27)(15 32 21 26)(16 31 22 25)(17 30 23 36)(18 29 24 35)(37 55 43 49)(38 54 44 60)(39 53 45 59)(40 52 46 58)(41 51 47 57)(42 50 48 56)(61 91 67 85)(62 90 68 96)(63 89 69 95)(64 88 70 94)(65 87 71 93)(66 86 72 92)
(1 31 90 38)(2 26 91 45)(3 33 92 40)(4 28 93 47)(5 35 94 42)(6 30 95 37)(7 25 96 44)(8 32 85 39)(9 27 86 46)(10 34 87 41)(11 29 88 48)(12 36 89 43)(13 68 57 75)(14 63 58 82)(15 70 59 77)(16 65 60 84)(17 72 49 79)(18 67 50 74)(19 62 51 81)(20 69 52 76)(21 64 53 83)(22 71 54 78)(23 66 55 73)(24 61 56 80)
(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 49)(24 50)(25 38)(26 39)(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 37)
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), (1,75,7,81)(2,74,8,80)(3,73,9,79)(4,84,10,78)(5,83,11,77)(6,82,12,76)(13,34,19,28)(14,33,20,27)(15,32,21,26)(16,31,22,25)(17,30,23,36)(18,29,24,35)(37,55,43,49)(38,54,44,60)(39,53,45,59)(40,52,46,58)(41,51,47,57)(42,50,48,56)(61,91,67,85)(62,90,68,96)(63,89,69,95)(64,88,70,94)(65,87,71,93)(66,86,72,92), (1,31,90,38)(2,26,91,45)(3,33,92,40)(4,28,93,47)(5,35,94,42)(6,30,95,37)(7,25,96,44)(8,32,85,39)(9,27,86,46)(10,34,87,41)(11,29,88,48)(12,36,89,43)(13,68,57,75)(14,63,58,82)(15,70,59,77)(16,65,60,84)(17,72,49,79)(18,67,50,74)(19,62,51,81)(20,69,52,76)(21,64,53,83)(22,71,54,78)(23,66,55,73)(24,61,56,80), (13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,49)(24,50)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,37)>;
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), (1,75,7,81)(2,74,8,80)(3,73,9,79)(4,84,10,78)(5,83,11,77)(6,82,12,76)(13,34,19,28)(14,33,20,27)(15,32,21,26)(16,31,22,25)(17,30,23,36)(18,29,24,35)(37,55,43,49)(38,54,44,60)(39,53,45,59)(40,52,46,58)(41,51,47,57)(42,50,48,56)(61,91,67,85)(62,90,68,96)(63,89,69,95)(64,88,70,94)(65,87,71,93)(66,86,72,92), (1,31,90,38)(2,26,91,45)(3,33,92,40)(4,28,93,47)(5,35,94,42)(6,30,95,37)(7,25,96,44)(8,32,85,39)(9,27,86,46)(10,34,87,41)(11,29,88,48)(12,36,89,43)(13,68,57,75)(14,63,58,82)(15,70,59,77)(16,65,60,84)(17,72,49,79)(18,67,50,74)(19,62,51,81)(20,69,52,76)(21,64,53,83)(22,71,54,78)(23,66,55,73)(24,61,56,80), (13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,49)(24,50)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,37) );
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)], [(1,75,7,81),(2,74,8,80),(3,73,9,79),(4,84,10,78),(5,83,11,77),(6,82,12,76),(13,34,19,28),(14,33,20,27),(15,32,21,26),(16,31,22,25),(17,30,23,36),(18,29,24,35),(37,55,43,49),(38,54,44,60),(39,53,45,59),(40,52,46,58),(41,51,47,57),(42,50,48,56),(61,91,67,85),(62,90,68,96),(63,89,69,95),(64,88,70,94),(65,87,71,93),(66,86,72,92)], [(1,31,90,38),(2,26,91,45),(3,33,92,40),(4,28,93,47),(5,35,94,42),(6,30,95,37),(7,25,96,44),(8,32,85,39),(9,27,86,46),(10,34,87,41),(11,29,88,48),(12,36,89,43),(13,68,57,75),(14,63,58,82),(15,70,59,77),(16,65,60,84),(17,72,49,79),(18,67,50,74),(19,62,51,81),(20,69,52,76),(21,64,53,83),(22,71,54,78),(23,66,55,73),(24,61,56,80)], [(13,51),(14,52),(15,53),(16,54),(17,55),(18,56),(19,57),(20,58),(21,59),(22,60),(23,49),(24,50),(25,38),(26,39),(27,40),(28,41),(29,42),(30,43),(31,44),(32,45),(33,46),(34,47),(35,48),(36,37)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | Q16 | C3⋊D4 | C3⋊D4 | C8.C22 | S3×D4 | C3⋊Q16 | Q8.14D6 |
| kernel | Dic6.37D4 | C6.SD16 | C12.55D4 | C2×C3⋊Q16 | C3×C22⋊Q8 | C22×Dic6 | C22⋊Q8 | Dic6 | C2×C12 | C22×C6 | C4⋊C4 | C22×C4 | C2×Q8 | C2×C6 | C2×C4 | C23 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 2 | 2 |
Matrix representation of Dic6.37D4 ►in GL6(𝔽73)
| 72 | 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 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 11 | 60 | 0 | 0 | 0 | 0 |
| 71 | 62 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 66 |
| 0 | 0 | 0 | 0 | 66 | 13 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 56 | 70 | 0 | 0 |
| 0 | 0 | 48 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 50 | 45 |
| 0 | 0 | 0 | 0 | 45 | 23 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 13 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(73))| [72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[11,71,0,0,0,0,60,62,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,60,66,0,0,0,0,66,13],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,56,48,0,0,0,0,70,17,0,0,0,0,0,0,50,45,0,0,0,0,45,23],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,13,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
Dic6.37D4 in GAP, Magma, Sage, TeX
{\rm Dic}_6._{37}D_4 % in TeX
G:=Group("Dic6.37D4"); // GroupNames label
G:=SmallGroup(192,609);
// by ID
G=gap.SmallGroup(192,609);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,254,219,184,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^4=d^2=1,b^2=a^6,b*a*b^-1=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^9*b,b*d=d*b,d*c*d=a^6*c^-1>;
// generators/relations