metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.16D4, C12.12D8, Q16.2Dic3, (C2×C8).50D6, C24.26(C2×C4), (C3×Q16).2C4, (C2×Q16).5S3, (C6×Q16).1C2, C8.2(C2×Dic3), C4.15(D4⋊S3), (C2×C12).117D4, C3⋊3(C8.17D4), C8.26(C3⋊D4), (C2×C6).31SD16, C12.C8.1C2, C24.C4.2C2, (C2×C24).30C22, C6.29(D4⋊C4), C12.16(C22⋊C4), C2.9(D4⋊Dic3), C4.4(C6.D4), C22.7(D4.S3), (C2×C4).25(C3⋊D4), SmallGroup(192,124)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q16.Dic3
G = < a,b,c,d | a8=1, b2=c6=a4, d2=a4c3, bab-1=a-1, ac=ca, dad-1=a3, cbc-1=a4b, dbd-1=a3b, dcd-1=c5 >
Character table of Q16.Dic3
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 12A | 12B | 12C | 12D | 12E | 12F | 16A | 16B | 16C | 16D | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 4 | 24 | 24 | 4 | 4 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -i | i | -1 | 1 | -1 | -1 | 1 | 1 | -i | i | -i | i | 1 | -1 | 1 | -1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | -i | i | -1 | 1 | 1 | 1 | -1 | -1 | i | -i | i | -i | 1 | -1 | 1 | -1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | i | -i | -1 | 1 | -1 | -1 | 1 | 1 | i | -i | i | -i | 1 | -1 | 1 | -1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | i | -i | -1 | 1 | 1 | 1 | -1 | -1 | -i | i | -i | i | 1 | -1 | 1 | -1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | -1 | 2 | 2 | -2 | -2 | -1 | -1 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | -2 | -2 | 2 | 2 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ13 | 2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -√2 | -√2 | √2 | √2 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ14 | 2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | √2 | √2 | -√2 | -√2 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ15 | 2 | 2 | -2 | -1 | -2 | 2 | -2 | 2 | 1 | 1 | -1 | -2 | -2 | 2 | 0 | 0 | 1 | -1 | 1 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ16 | 2 | 2 | -2 | -1 | -2 | 2 | 2 | -2 | 1 | 1 | -1 | -2 | -2 | 2 | 0 | 0 | 1 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ17 | 2 | 2 | -2 | -1 | -2 | 2 | 0 | 0 | 1 | 1 | -1 | 2 | 2 | -2 | 0 | 0 | 1 | -1 | √-3 | -√-3 | √-3 | -√-3 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -2 | -2 | -2 | 0 | 0 | -1 | -1 | √-3 | -√-3 | -√-3 | √-3 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ19 | 2 | 2 | -2 | -1 | -2 | 2 | 0 | 0 | 1 | 1 | -1 | 2 | 2 | -2 | 0 | 0 | 1 | -1 | -√-3 | √-3 | -√-3 | √-3 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | complex lifted from C3⋊D4 |
| ρ20 | 2 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -2 | -2 | -2 | 0 | 0 | -1 | -1 | -√-3 | √-3 | √-3 | -√-3 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ21 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | √-2 | -√-2 | -√-2 | √-2 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ22 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | -√-2 | √-2 | √-2 | -√-2 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ23 | 4 | 4 | -4 | -2 | 4 | -4 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4⋊S3, Schur index 2 |
| ρ24 | 4 | 4 | 4 | -2 | -4 | -4 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4.S3, Schur index 2 |
| ρ25 | 4 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | 0 | -2√2 | symplectic lifted from C8.17D4, Schur index 2 |
| ρ26 | 4 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 0 | 2√2 | symplectic lifted from C8.17D4, Schur index 2 |
| ρ27 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 2√-3 | -2√-3 | 2 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-6 | √2 | √-6 | -√2 | complex faithful |
| ρ28 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | -2√-3 | 2√-3 | 2 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-6 | -√2 | √-6 | √2 | complex faithful |
| ρ29 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | -2√-3 | 2√-3 | 2 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-6 | √2 | -√-6 | -√2 | complex faithful |
| ρ30 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 2√-3 | -2√-3 | 2 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-6 | -√2 | -√-6 | √2 | complex faithful |
►On 96 points
Generators in S96
(1 15 4 18 7 21 10 24)(2 16 5 19 8 22 11 13)(3 17 6 20 9 23 12 14)(25 44 28 47 31 38 34 41)(26 45 29 48 32 39 35 42)(27 46 30 37 33 40 36 43)(49 61 58 70 55 67 52 64)(50 62 59 71 56 68 53 65)(51 63 60 72 57 69 54 66)(73 88 82 85 79 94 76 91)(74 89 83 86 80 95 77 92)(75 90 84 87 81 96 78 93)
(1 34 7 28)(2 29 8 35)(3 36 9 30)(4 31 10 25)(5 26 11 32)(6 33 12 27)(13 48 19 42)(14 43 20 37)(15 38 21 44)(16 45 22 39)(17 40 23 46)(18 47 24 41)(49 85 55 91)(50 92 56 86)(51 87 57 93)(52 94 58 88)(53 89 59 95)(54 96 60 90)(61 82 67 76)(62 77 68 83)(63 84 69 78)(64 79 70 73)(65 74 71 80)(66 81 72 75)
(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 58 10 55 7 52 4 49)(2 51 11 60 8 57 5 54)(3 56 12 53 9 50 6 59)(13 69 22 66 19 63 16 72)(14 62 23 71 20 68 17 65)(15 67 24 64 21 61 18 70)(25 73 34 82 31 79 28 76)(26 78 35 75 32 84 29 81)(27 83 36 80 33 77 30 74)(37 86 46 95 43 92 40 89)(38 91 47 88 44 85 41 94)(39 96 48 93 45 90 42 87)
G:=sub<Sym(96)| (1,15,4,18,7,21,10,24)(2,16,5,19,8,22,11,13)(3,17,6,20,9,23,12,14)(25,44,28,47,31,38,34,41)(26,45,29,48,32,39,35,42)(27,46,30,37,33,40,36,43)(49,61,58,70,55,67,52,64)(50,62,59,71,56,68,53,65)(51,63,60,72,57,69,54,66)(73,88,82,85,79,94,76,91)(74,89,83,86,80,95,77,92)(75,90,84,87,81,96,78,93), (1,34,7,28)(2,29,8,35)(3,36,9,30)(4,31,10,25)(5,26,11,32)(6,33,12,27)(13,48,19,42)(14,43,20,37)(15,38,21,44)(16,45,22,39)(17,40,23,46)(18,47,24,41)(49,85,55,91)(50,92,56,86)(51,87,57,93)(52,94,58,88)(53,89,59,95)(54,96,60,90)(61,82,67,76)(62,77,68,83)(63,84,69,78)(64,79,70,73)(65,74,71,80)(66,81,72,75), (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), (1,58,10,55,7,52,4,49)(2,51,11,60,8,57,5,54)(3,56,12,53,9,50,6,59)(13,69,22,66,19,63,16,72)(14,62,23,71,20,68,17,65)(15,67,24,64,21,61,18,70)(25,73,34,82,31,79,28,76)(26,78,35,75,32,84,29,81)(27,83,36,80,33,77,30,74)(37,86,46,95,43,92,40,89)(38,91,47,88,44,85,41,94)(39,96,48,93,45,90,42,87)>;
G:=Group( (1,15,4,18,7,21,10,24)(2,16,5,19,8,22,11,13)(3,17,6,20,9,23,12,14)(25,44,28,47,31,38,34,41)(26,45,29,48,32,39,35,42)(27,46,30,37,33,40,36,43)(49,61,58,70,55,67,52,64)(50,62,59,71,56,68,53,65)(51,63,60,72,57,69,54,66)(73,88,82,85,79,94,76,91)(74,89,83,86,80,95,77,92)(75,90,84,87,81,96,78,93), (1,34,7,28)(2,29,8,35)(3,36,9,30)(4,31,10,25)(5,26,11,32)(6,33,12,27)(13,48,19,42)(14,43,20,37)(15,38,21,44)(16,45,22,39)(17,40,23,46)(18,47,24,41)(49,85,55,91)(50,92,56,86)(51,87,57,93)(52,94,58,88)(53,89,59,95)(54,96,60,90)(61,82,67,76)(62,77,68,83)(63,84,69,78)(64,79,70,73)(65,74,71,80)(66,81,72,75), (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), (1,58,10,55,7,52,4,49)(2,51,11,60,8,57,5,54)(3,56,12,53,9,50,6,59)(13,69,22,66,19,63,16,72)(14,62,23,71,20,68,17,65)(15,67,24,64,21,61,18,70)(25,73,34,82,31,79,28,76)(26,78,35,75,32,84,29,81)(27,83,36,80,33,77,30,74)(37,86,46,95,43,92,40,89)(38,91,47,88,44,85,41,94)(39,96,48,93,45,90,42,87) );
G=PermutationGroup([[(1,15,4,18,7,21,10,24),(2,16,5,19,8,22,11,13),(3,17,6,20,9,23,12,14),(25,44,28,47,31,38,34,41),(26,45,29,48,32,39,35,42),(27,46,30,37,33,40,36,43),(49,61,58,70,55,67,52,64),(50,62,59,71,56,68,53,65),(51,63,60,72,57,69,54,66),(73,88,82,85,79,94,76,91),(74,89,83,86,80,95,77,92),(75,90,84,87,81,96,78,93)], [(1,34,7,28),(2,29,8,35),(3,36,9,30),(4,31,10,25),(5,26,11,32),(6,33,12,27),(13,48,19,42),(14,43,20,37),(15,38,21,44),(16,45,22,39),(17,40,23,46),(18,47,24,41),(49,85,55,91),(50,92,56,86),(51,87,57,93),(52,94,58,88),(53,89,59,95),(54,96,60,90),(61,82,67,76),(62,77,68,83),(63,84,69,78),(64,79,70,73),(65,74,71,80),(66,81,72,75)], [(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)], [(1,58,10,55,7,52,4,49),(2,51,11,60,8,57,5,54),(3,56,12,53,9,50,6,59),(13,69,22,66,19,63,16,72),(14,62,23,71,20,68,17,65),(15,67,24,64,21,61,18,70),(25,73,34,82,31,79,28,76),(26,78,35,75,32,84,29,81),(27,83,36,80,33,77,30,74),(37,86,46,95,43,92,40,89),(38,91,47,88,44,85,41,94),(39,96,48,93,45,90,42,87)]])
Matrix representation of Q16.Dic3 ►in GL4(𝔽7) generated by
| 2 | 6 | 0 | 6 |
| 5 | 2 | 2 | 6 |
| 0 | 4 | 0 | 2 |
| 1 | 5 | 6 | 3 |
| 0 | 0 | 1 | 2 |
| 0 | 2 | 1 | 0 |
| 0 | 2 | 5 | 0 |
| 3 | 6 | 1 | 0 |
| 0 | 3 | 1 | 0 |
| 0 | 3 | 0 | 4 |
| 2 | 6 | 0 | 6 |
| 6 | 0 | 3 | 4 |
| 4 | 4 | 3 | 2 |
| 0 | 6 | 4 | 4 |
| 0 | 2 | 3 | 1 |
| 1 | 4 | 6 | 1 |
G:=sub<GL(4,GF(7))| [2,5,0,1,6,2,4,5,0,2,0,6,6,6,2,3],[0,0,0,3,0,2,2,6,1,1,5,1,2,0,0,0],[0,0,2,6,3,3,6,0,1,0,0,3,0,4,6,4],[4,0,0,1,4,6,2,4,3,4,3,6,2,4,1,1] >;
Q16.Dic3 in GAP, Magma, Sage, TeX
Q_{16}.{\rm Dic}_3 % in TeX
G:=Group("Q16.Dic3"); // GroupNames label
G:=SmallGroup(192,124);
// by ID
G=gap.SmallGroup(192,124);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,232,387,184,675,794,80,1684,851,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^8=1,b^2=c^6=a^4,d^2=a^4*c^3,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^3,c*b*c^-1=a^4*b,d*b*d^-1=a^3*b,d*c*d^-1=c^5>;
// generators/relations
Export
Subgroup lattice of Q16.Dic3 in TeX
Character table of Q16.Dic3 in TeX