metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8.9D6, C24.33D4, C12.52D8, Q16.10D6, C24.31C23, Dic12.14C22, D8.S3⋊6C2, C4○D8.3S3, C3⋊Q32⋊6C2, (C2×C8).99D6, (C2×C6).11D8, C6.70(C2×D8), C8.8(C3⋊D4), C3⋊5(Q32⋊C2), C3⋊C16.4C22, C12.C8⋊7C2, C4.25(D4⋊S3), C12.193(C2×D4), (C2×C12).187D4, C8.37(C22×S3), (C3×D8).9C22, (C2×Dic12)⋊22C2, C22.6(D4⋊S3), (C2×C24).105C22, (C3×Q16).10C22, C2.25(C2×D4⋊S3), (C3×C4○D8).4C2, C4.19(C2×C3⋊D4), (C2×C4).82(C3⋊D4), SmallGroup(192,754)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8.9D6
G = < a,b,c,d | a8=b2=c6=1, d2=a4, bab=dad-1=a-1, ac=ca, cbc-1=a4b, dbd-1=ab, dcd-1=c-1 >
Subgroups: 232 in 82 conjugacy classes, 35 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, Q8, Dic3, C12, C12, C2×C6, C2×C6, C16, C2×C8, D8, SD16, Q16, Q16, C2×Q8, C4○D4, C24, Dic6, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×Q8, M5(2), SD32, Q32, C2×Q16, C4○D8, C3⋊C16, Dic12, Dic12, C2×C24, C3×D8, C3×SD16, C3×Q16, C2×Dic6, C3×C4○D4, Q32⋊C2, C12.C8, D8.S3, C3⋊Q32, C2×Dic12, C3×C4○D8, D8.9D6
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C3⋊D4, C22×S3, C2×D8, D4⋊S3, C2×C3⋊D4, Q32⋊C2, C2×D4⋊S3, D8.9D6
Character table of D8.9D6
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 12A | 12B | 12C | 12D | 12E | 16A | 16B | 16C | 16D | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 2 | 8 | 2 | 2 | 2 | 8 | 24 | 24 | 2 | 4 | 8 | 8 | 2 | 2 | 4 | 2 | 2 | 4 | 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 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ6 | 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 |
| ρ7 | 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 |
| ρ8 | 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 |
| ρ9 | 2 | 2 | 2 | -2 | -1 | 2 | 2 | -2 | 0 | 0 | -1 | -1 | 1 | 1 | 2 | 2 | 2 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | -2 | -2 | -1 | -2 | 2 | 2 | 0 | 0 | -1 | 1 | 1 | 1 | 2 | 2 | -2 | 1 | 1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ13 | 2 | 2 | -2 | 2 | -1 | -2 | 2 | -2 | 0 | 0 | -1 | 1 | -1 | -1 | 2 | 2 | -2 | 1 | 1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | -2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | -2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | √2 | √2 | -√2 | -√2 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ16 | 2 | 2 | -2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | √2 | -√2 | √2 | -√2 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ17 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | -√2 | -√2 | √2 | √2 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ18 | 2 | 2 | -2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | -√2 | √2 | -√2 | √2 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ19 | 2 | 2 | -2 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | -1 | 1 | -√-3 | √-3 | -2 | -2 | 2 | 1 | 1 | -1 | -√-3 | √-3 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | complex lifted from C3⋊D4 |
| ρ20 | 2 | 2 | -2 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | -1 | 1 | √-3 | -√-3 | -2 | -2 | 2 | 1 | 1 | -1 | √-3 | -√-3 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | complex lifted from C3⋊D4 |
| ρ21 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -√-3 | √-3 | -2 | -2 | -2 | -1 | -1 | -1 | √-3 | -√-3 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ22 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | √-3 | -√-3 | -2 | -2 | -2 | -1 | -1 | -1 | -√-3 | √-3 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ23 | 4 | 4 | -4 | 0 | -2 | 4 | -4 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4⋊S3, Schur index 2 |
| ρ24 | 4 | 4 | 4 | 0 | -2 | -4 | -4 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4⋊S3, Schur index 2 |
| ρ25 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 0 | 2√2 | symplectic lifted from Q32⋊C2, Schur index 2 |
| ρ26 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | 0 | -2√2 | symplectic lifted from Q32⋊C2, Schur index 2 |
| ρ27 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | -2√3 | 2√3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √6 | √2 | -√6 | -√2 | symplectic faithful, Schur index 2 |
| ρ28 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 2√3 | -2√3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√6 | √2 | √6 | -√2 | symplectic faithful, Schur index 2 |
| ρ29 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 2√3 | -2√3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √6 | -√2 | -√6 | √2 | symplectic faithful, Schur index 2 |
| ρ30 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | -2√3 | 2√3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√6 | -√2 | √6 | √2 | symplectic faithful, Schur index 2 |
►On 96 points
Generators in S96
(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 8)(2 7)(3 6)(4 5)(9 15)(10 14)(11 13)(18 24)(19 23)(20 22)(25 30)(26 29)(27 28)(31 32)(33 36)(34 35)(37 40)(38 39)(41 46)(42 45)(43 44)(47 48)(49 52)(50 51)(53 56)(54 55)(57 62)(58 61)(59 60)(63 64)(66 72)(67 71)(68 70)(73 75)(76 80)(77 79)(81 85)(82 84)(86 88)(89 93)(90 92)(94 96)
(1 46 39 62 32 49)(2 47 40 63 25 50)(3 48 33 64 26 51)(4 41 34 57 27 52)(5 42 35 58 28 53)(6 43 36 59 29 54)(7 44 37 60 30 55)(8 45 38 61 31 56)(9 73 96 82 22 72)(10 74 89 83 23 65)(11 75 90 84 24 66)(12 76 91 85 17 67)(13 77 92 86 18 68)(14 78 93 87 19 69)(15 79 94 88 20 70)(16 80 95 81 21 71)
(1 94 5 90)(2 93 6 89)(3 92 7 96)(4 91 8 95)(9 33 13 37)(10 40 14 36)(11 39 15 35)(12 38 16 34)(17 31 21 27)(18 30 22 26)(19 29 23 25)(20 28 24 32)(41 76 45 80)(42 75 46 79)(43 74 47 78)(44 73 48 77)(49 88 53 84)(50 87 54 83)(51 86 55 82)(52 85 56 81)(57 67 61 71)(58 66 62 70)(59 65 63 69)(60 72 64 68)
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), (1,8)(2,7)(3,6)(4,5)(9,15)(10,14)(11,13)(18,24)(19,23)(20,22)(25,30)(26,29)(27,28)(31,32)(33,36)(34,35)(37,40)(38,39)(41,46)(42,45)(43,44)(47,48)(49,52)(50,51)(53,56)(54,55)(57,62)(58,61)(59,60)(63,64)(66,72)(67,71)(68,70)(73,75)(76,80)(77,79)(81,85)(82,84)(86,88)(89,93)(90,92)(94,96), (1,46,39,62,32,49)(2,47,40,63,25,50)(3,48,33,64,26,51)(4,41,34,57,27,52)(5,42,35,58,28,53)(6,43,36,59,29,54)(7,44,37,60,30,55)(8,45,38,61,31,56)(9,73,96,82,22,72)(10,74,89,83,23,65)(11,75,90,84,24,66)(12,76,91,85,17,67)(13,77,92,86,18,68)(14,78,93,87,19,69)(15,79,94,88,20,70)(16,80,95,81,21,71), (1,94,5,90)(2,93,6,89)(3,92,7,96)(4,91,8,95)(9,33,13,37)(10,40,14,36)(11,39,15,35)(12,38,16,34)(17,31,21,27)(18,30,22,26)(19,29,23,25)(20,28,24,32)(41,76,45,80)(42,75,46,79)(43,74,47,78)(44,73,48,77)(49,88,53,84)(50,87,54,83)(51,86,55,82)(52,85,56,81)(57,67,61,71)(58,66,62,70)(59,65,63,69)(60,72,64,68)>;
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), (1,8)(2,7)(3,6)(4,5)(9,15)(10,14)(11,13)(18,24)(19,23)(20,22)(25,30)(26,29)(27,28)(31,32)(33,36)(34,35)(37,40)(38,39)(41,46)(42,45)(43,44)(47,48)(49,52)(50,51)(53,56)(54,55)(57,62)(58,61)(59,60)(63,64)(66,72)(67,71)(68,70)(73,75)(76,80)(77,79)(81,85)(82,84)(86,88)(89,93)(90,92)(94,96), (1,46,39,62,32,49)(2,47,40,63,25,50)(3,48,33,64,26,51)(4,41,34,57,27,52)(5,42,35,58,28,53)(6,43,36,59,29,54)(7,44,37,60,30,55)(8,45,38,61,31,56)(9,73,96,82,22,72)(10,74,89,83,23,65)(11,75,90,84,24,66)(12,76,91,85,17,67)(13,77,92,86,18,68)(14,78,93,87,19,69)(15,79,94,88,20,70)(16,80,95,81,21,71), (1,94,5,90)(2,93,6,89)(3,92,7,96)(4,91,8,95)(9,33,13,37)(10,40,14,36)(11,39,15,35)(12,38,16,34)(17,31,21,27)(18,30,22,26)(19,29,23,25)(20,28,24,32)(41,76,45,80)(42,75,46,79)(43,74,47,78)(44,73,48,77)(49,88,53,84)(50,87,54,83)(51,86,55,82)(52,85,56,81)(57,67,61,71)(58,66,62,70)(59,65,63,69)(60,72,64,68) );
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)], [(1,8),(2,7),(3,6),(4,5),(9,15),(10,14),(11,13),(18,24),(19,23),(20,22),(25,30),(26,29),(27,28),(31,32),(33,36),(34,35),(37,40),(38,39),(41,46),(42,45),(43,44),(47,48),(49,52),(50,51),(53,56),(54,55),(57,62),(58,61),(59,60),(63,64),(66,72),(67,71),(68,70),(73,75),(76,80),(77,79),(81,85),(82,84),(86,88),(89,93),(90,92),(94,96)], [(1,46,39,62,32,49),(2,47,40,63,25,50),(3,48,33,64,26,51),(4,41,34,57,27,52),(5,42,35,58,28,53),(6,43,36,59,29,54),(7,44,37,60,30,55),(8,45,38,61,31,56),(9,73,96,82,22,72),(10,74,89,83,23,65),(11,75,90,84,24,66),(12,76,91,85,17,67),(13,77,92,86,18,68),(14,78,93,87,19,69),(15,79,94,88,20,70),(16,80,95,81,21,71)], [(1,94,5,90),(2,93,6,89),(3,92,7,96),(4,91,8,95),(9,33,13,37),(10,40,14,36),(11,39,15,35),(12,38,16,34),(17,31,21,27),(18,30,22,26),(19,29,23,25),(20,28,24,32),(41,76,45,80),(42,75,46,79),(43,74,47,78),(44,73,48,77),(49,88,53,84),(50,87,54,83),(51,86,55,82),(52,85,56,81),(57,67,61,71),(58,66,62,70),(59,65,63,69),(60,72,64,68)]])
Matrix representation of D8.9D6 ►in GL4(𝔽97) generated by
| 7 | 0 | 0 | 90 |
| 46 | 14 | 94 | 46 |
| 53 | 65 | 0 | 0 |
| 7 | 0 | 0 | 7 |
| 7 | 0 | 0 | 7 |
| 51 | 0 | 3 | 51 |
| 53 | 65 | 0 | 0 |
| 7 | 0 | 0 | 90 |
| 52 | 29 | 43 | 81 |
| 90 | 39 | 11 | 22 |
| 10 | 69 | 6 | 0 |
| 68 | 29 | 0 | 0 |
| 73 | 61 | 86 | 89 |
| 74 | 16 | 48 | 94 |
| 21 | 75 | 82 | 37 |
| 74 | 42 | 68 | 23 |
G:=sub<GL(4,GF(97))| [7,46,53,7,0,14,65,0,0,94,0,0,90,46,0,7],[7,51,53,7,0,0,65,0,0,3,0,0,7,51,0,90],[52,90,10,68,29,39,69,29,43,11,6,0,81,22,0,0],[73,74,21,74,61,16,75,42,86,48,82,68,89,94,37,23] >;
D8.9D6 in GAP, Magma, Sage, TeX
D_8._9D_6
% in TeX
G:=Group("D8.9D6"); // GroupNames label
G:=SmallGroup(192,754);
// by ID
G=gap.SmallGroup(192,754);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,254,387,675,185,192,1684,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^6=1,d^2=a^4,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^4*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations
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