metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6.11D4, C4.91(S3×D4), C4⋊C4.146D6, Q8⋊C4⋊9S3, (C2×C8).120D6, (C2×Q8).37D6, Dic3⋊C8⋊13C2, C4.6(C4○D12), C6.45(C4○D8), C12.115(C2×D4), C3⋊2(Q8.D4), C6.D8.2C2, Dic6⋊C4⋊5C2, C12.17(C4○D4), C6.22(C4⋊D4), (C2×Dic3).28D4, C22.192(S3×D4), (C6×Q8).21C22, C2.25(Dic3⋊D4), (C2×C24).131C22, (C2×C12).238C23, C12.23D4.3C2, C2.11(Q16⋊S3), (C2×D12).58C22, C6.56(C8.C22), C2.14(Q8.7D6), (C2×Dic6).67C22, (C4×Dic3).22C22, (C2×C3⋊Q16)⋊2C2, (C2×C24⋊C2).3C2, (C3×Q8⋊C4)⋊9C2, (C2×C6).251(C2×D4), (C2×C3⋊C8).33C22, (C3×C4⋊C4).39C22, (C2×C4).345(C22×S3), SmallGroup(192,357)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6.11D4
G = < a,b,c,d | a12=c4=d2=1, b2=a6, bab-1=dad=a-1, cac-1=a5, bc=cb, dbd=a3b, dcd=a6c-1 >
Subgroups: 344 in 112 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C2×D4, C2×Q8, C2×Q8, C3⋊C8, C24, Dic6, Dic6, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, D4⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4.4D4, C2×SD16, C2×Q16, C24⋊C2, C2×C3⋊C8, C4×Dic3, C4×Dic3, Dic3⋊C4, D6⋊C4, C3⋊Q16, C3×C4⋊C4, C2×C24, C2×Dic6, C2×D12, C6×Q8, Q8.D4, C6.D8, Dic3⋊C8, C3×Q8⋊C4, Dic6⋊C4, C2×C24⋊C2, C2×C3⋊Q16, C12.23D4, Dic6.11D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C22×S3, C4⋊D4, C4○D8, C8.C22, C4○D12, S3×D4, Q8.D4, Dic3⋊D4, Q8.7D6, Q16⋊S3, Dic6.11D4
►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 57 7 51)(2 56 8 50)(3 55 9 49)(4 54 10 60)(5 53 11 59)(6 52 12 58)(13 82 19 76)(14 81 20 75)(15 80 21 74)(16 79 22 73)(17 78 23 84)(18 77 24 83)(25 63 31 69)(26 62 32 68)(27 61 33 67)(28 72 34 66)(29 71 35 65)(30 70 36 64)(37 92 43 86)(38 91 44 85)(39 90 45 96)(40 89 46 95)(41 88 47 94)(42 87 48 93)
(1 90 21 27)(2 95 22 32)(3 88 23 25)(4 93 24 30)(5 86 13 35)(6 91 14 28)(7 96 15 33)(8 89 16 26)(9 94 17 31)(10 87 18 36)(11 92 19 29)(12 85 20 34)(37 82 65 53)(38 75 66 58)(39 80 67 51)(40 73 68 56)(41 78 69 49)(42 83 70 54)(43 76 71 59)(44 81 72 52)(45 74 61 57)(46 79 62 50)(47 84 63 55)(48 77 64 60)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 17)(14 16)(18 24)(19 23)(20 22)(25 86)(26 85)(27 96)(28 95)(29 94)(30 93)(31 92)(32 91)(33 90)(34 89)(35 88)(36 87)(37 72)(38 71)(39 70)(40 69)(41 68)(42 67)(43 66)(44 65)(45 64)(46 63)(47 62)(48 61)(49 50)(51 60)(52 59)(53 58)(54 57)(55 56)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)
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), (1,57,7,51)(2,56,8,50)(3,55,9,49)(4,54,10,60)(5,53,11,59)(6,52,12,58)(13,82,19,76)(14,81,20,75)(15,80,21,74)(16,79,22,73)(17,78,23,84)(18,77,24,83)(25,63,31,69)(26,62,32,68)(27,61,33,67)(28,72,34,66)(29,71,35,65)(30,70,36,64)(37,92,43,86)(38,91,44,85)(39,90,45,96)(40,89,46,95)(41,88,47,94)(42,87,48,93), (1,90,21,27)(2,95,22,32)(3,88,23,25)(4,93,24,30)(5,86,13,35)(6,91,14,28)(7,96,15,33)(8,89,16,26)(9,94,17,31)(10,87,18,36)(11,92,19,29)(12,85,20,34)(37,82,65,53)(38,75,66,58)(39,80,67,51)(40,73,68,56)(41,78,69,49)(42,83,70,54)(43,76,71,59)(44,81,72,52)(45,74,61,57)(46,79,62,50)(47,84,63,55)(48,77,64,60), (2,12)(3,11)(4,10)(5,9)(6,8)(13,17)(14,16)(18,24)(19,23)(20,22)(25,86)(26,85)(27,96)(28,95)(29,94)(30,93)(31,92)(32,91)(33,90)(34,89)(35,88)(36,87)(37,72)(38,71)(39,70)(40,69)(41,68)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,50)(51,60)(52,59)(53,58)(54,57)(55,56)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)>;
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), (1,57,7,51)(2,56,8,50)(3,55,9,49)(4,54,10,60)(5,53,11,59)(6,52,12,58)(13,82,19,76)(14,81,20,75)(15,80,21,74)(16,79,22,73)(17,78,23,84)(18,77,24,83)(25,63,31,69)(26,62,32,68)(27,61,33,67)(28,72,34,66)(29,71,35,65)(30,70,36,64)(37,92,43,86)(38,91,44,85)(39,90,45,96)(40,89,46,95)(41,88,47,94)(42,87,48,93), (1,90,21,27)(2,95,22,32)(3,88,23,25)(4,93,24,30)(5,86,13,35)(6,91,14,28)(7,96,15,33)(8,89,16,26)(9,94,17,31)(10,87,18,36)(11,92,19,29)(12,85,20,34)(37,82,65,53)(38,75,66,58)(39,80,67,51)(40,73,68,56)(41,78,69,49)(42,83,70,54)(43,76,71,59)(44,81,72,52)(45,74,61,57)(46,79,62,50)(47,84,63,55)(48,77,64,60), (2,12)(3,11)(4,10)(5,9)(6,8)(13,17)(14,16)(18,24)(19,23)(20,22)(25,86)(26,85)(27,96)(28,95)(29,94)(30,93)(31,92)(32,91)(33,90)(34,89)(35,88)(36,87)(37,72)(38,71)(39,70)(40,69)(41,68)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,50)(51,60)(52,59)(53,58)(54,57)(55,56)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79) );
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)], [(1,57,7,51),(2,56,8,50),(3,55,9,49),(4,54,10,60),(5,53,11,59),(6,52,12,58),(13,82,19,76),(14,81,20,75),(15,80,21,74),(16,79,22,73),(17,78,23,84),(18,77,24,83),(25,63,31,69),(26,62,32,68),(27,61,33,67),(28,72,34,66),(29,71,35,65),(30,70,36,64),(37,92,43,86),(38,91,44,85),(39,90,45,96),(40,89,46,95),(41,88,47,94),(42,87,48,93)], [(1,90,21,27),(2,95,22,32),(3,88,23,25),(4,93,24,30),(5,86,13,35),(6,91,14,28),(7,96,15,33),(8,89,16,26),(9,94,17,31),(10,87,18,36),(11,92,19,29),(12,85,20,34),(37,82,65,53),(38,75,66,58),(39,80,67,51),(40,73,68,56),(41,78,69,49),(42,83,70,54),(43,76,71,59),(44,81,72,52),(45,74,61,57),(46,79,62,50),(47,84,63,55),(48,77,64,60)], [(2,12),(3,11),(4,10),(5,9),(6,8),(13,17),(14,16),(18,24),(19,23),(20,22),(25,86),(26,85),(27,96),(28,95),(29,94),(30,93),(31,92),(32,91),(33,90),(34,89),(35,88),(36,87),(37,72),(38,71),(39,70),(40,69),(41,68),(42,67),(43,66),(44,65),(45,64),(46,63),(47,62),(48,61),(49,50),(51,60),(52,59),(53,58),(54,57),(55,56),(73,84),(74,83),(75,82),(76,81),(77,80),(78,79)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 24 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 8 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | C4○D4 | C4○D8 | C4○D12 | C8.C22 | S3×D4 | S3×D4 | Q8.7D6 | Q16⋊S3 |
| kernel | Dic6.11D4 | C6.D8 | Dic3⋊C8 | C3×Q8⋊C4 | Dic6⋊C4 | C2×C24⋊C2 | C2×C3⋊Q16 | C12.23D4 | Q8⋊C4 | Dic6 | C2×Dic3 | C4⋊C4 | C2×C8 | C2×Q8 | C12 | C6 | C4 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of Dic6.11D4 ►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 | 72 | 70 |
| 0 | 0 | 0 | 0 | 25 | 1 |
| 0 | 72 | 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 | 18 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 70 | 0 | 0 |
| 0 | 0 | 26 | 71 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 0 | 0 | 27 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 50 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 48 | 72 |
G:=sub<GL(6,GF(73))| [72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,25,0,0,0,0,70,1],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,18,0],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,2,26,0,0,0,0,70,71,0,0,0,0,0,0,27,0,0,0,0,0,0,27],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,50,0,0,0,0,0,72,0,0,0,0,0,0,1,48,0,0,0,0,0,72] >;
Dic6.11D4 in GAP, Magma, Sage, TeX
{\rm Dic}_6._{11}D_4 % in TeX
G:=Group("Dic6.11D4"); // GroupNames label
G:=SmallGroup(192,357);
// by ID
G=gap.SmallGroup(192,357);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,344,1094,135,184,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^4=d^2=1,b^2=a^6,b*a*b^-1=d*a*d=a^-1,c*a*c^-1=a^5,b*c=c*b,d*b*d=a^3*b,d*c*d=a^6*c^-1>;
// generators/relations