metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q32⋊2S3, D6.9D8, C16.3D6, Q16.4D6, C24.21C23, C48.10C22, Dic3.11D8, D24.4C22, Dic12.6C22, C3⋊C8.5D4, C4.9(S3×D4), D6.C8⋊4C2, (C3×Q32)⋊4C2, (S3×Q16)⋊5C2, C3⋊Q32⋊4C2, C48⋊C2⋊4C2, C6.40(C2×D8), C2.24(S3×D8), D24⋊C2.C2, (C4×S3).10D4, C8.6D6⋊3C2, C12.15(C2×D4), C3⋊3(Q32⋊C2), C3⋊C16.2C22, (S3×C8).6C22, C8.27(C22×S3), (C3×Q16).5C22, SmallGroup(192,477)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q32⋊S3
G = < a,b,c,d | a16=c3=d2=1, b2=a8, bab-1=a-1, ac=ca, dad=a9, bc=cb, dbd=a8b, dcd=c-1 >
Subgroups: 284 in 82 conjugacy classes, 31 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C8, C2×C4, D4, Q8, Dic3, Dic3, C12, C12, D6, D6, C16, C16, C2×C8, D8, SD16, Q16, Q16, C2×Q8, C4○D4, C3⋊C8, C24, Dic6, C4×S3, C4×S3, D12, C3×Q8, M5(2), SD32, Q32, Q32, C2×Q16, C4○D8, C3⋊C16, C48, S3×C8, D24, Dic12, Q8⋊2S3, C3⋊Q16, C3×Q16, S3×Q8, Q8⋊3S3, Q32⋊C2, D6.C8, C48⋊C2, C8.6D6, C3⋊Q32, C3×Q32, S3×Q16, D24⋊C2, Q32⋊S3
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C22×S3, C2×D8, S3×D4, Q32⋊C2, S3×D8, Q32⋊S3
Character table of Q32⋊S3
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 6 | 8A | 8B | 8C | 12A | 12B | 12C | 16A | 16B | 16C | 16D | 24A | 24B | 48A | 48B | 48C | 48D | |
| size | 1 | 1 | 6 | 24 | 2 | 2 | 6 | 8 | 8 | 24 | 2 | 2 | 2 | 12 | 4 | 16 | 16 | 4 | 4 | 12 | 12 | 4 | 4 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 0 | 0 | -1 | 2 | 0 | 2 | 2 | 0 | -1 | 2 | 2 | 0 | -1 | -1 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 0 | 0 | -1 | 2 | 0 | -2 | 2 | 0 | -1 | 2 | 2 | 0 | -1 | 1 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 0 | 0 | -1 | 2 | 0 | -2 | -2 | 0 | -1 | 2 | 2 | 0 | -1 | 1 | 1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | -2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 0 | 0 | -1 | 2 | 0 | 2 | -2 | 0 | -1 | 2 | 2 | 0 | -1 | -1 | 1 | -2 | -2 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ15 | 2 | 2 | -2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | √2 | -√2 | -√2 | √2 | 0 | 0 | -√2 | -√2 | √2 | √2 | orthogonal lifted from D8 |
| ρ16 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | √2 | -√2 | √2 | -√2 | 0 | 0 | -√2 | -√2 | √2 | √2 | orthogonal lifted from D8 |
| ρ17 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | -√2 | √2 | -√2 | √2 | 0 | 0 | √2 | √2 | -√2 | -√2 | orthogonal lifted from D8 |
| ρ18 | 2 | 2 | -2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | -√2 | √2 | √2 | -√2 | 0 | 0 | √2 | √2 | -√2 | -√2 | orthogonal lifted from D8 |
| ρ19 | 4 | 4 | 0 | 0 | -2 | 4 | 0 | 0 | 0 | 0 | -2 | -4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ20 | 4 | 4 | 0 | 0 | -2 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | √2 | √2 | -√2 | -√2 | orthogonal lifted from S3×D8 |
| ρ21 | 4 | 4 | 0 | 0 | -2 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | -√2 | -√2 | √2 | √2 | orthogonal lifted from S3×D8 |
| ρ22 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | symplectic lifted from Q32⋊C2, Schur index 2 |
| ρ23 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | symplectic lifted from Q32⋊C2, Schur index 2 |
| ρ24 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | -2ζ167ζ3-ζ167+2ζ16ζ3+ζ16 | 2ζ167ζ3+ζ167-2ζ16ζ3-ζ16 | 2ζ165ζ32+ζ165-2ζ163ζ32-ζ163 | 2ζ1613ζ32+ζ1613-2ζ1611ζ32-ζ1611 | complex faithful |
| ρ25 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | 2ζ1613ζ32+ζ1613-2ζ1611ζ32-ζ1611 | 2ζ165ζ32+ζ165-2ζ163ζ32-ζ163 | -2ζ167ζ3-ζ167+2ζ16ζ3+ζ16 | 2ζ167ζ3+ζ167-2ζ16ζ3-ζ16 | complex faithful |
| ρ26 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | 2ζ165ζ32+ζ165-2ζ163ζ32-ζ163 | 2ζ1613ζ32+ζ1613-2ζ1611ζ32-ζ1611 | 2ζ167ζ3+ζ167-2ζ16ζ3-ζ16 | -2ζ167ζ3-ζ167+2ζ16ζ3+ζ16 | complex faithful |
| ρ27 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | 2ζ167ζ3+ζ167-2ζ16ζ3-ζ16 | -2ζ167ζ3-ζ167+2ζ16ζ3+ζ16 | 2ζ1613ζ32+ζ1613-2ζ1611ζ32-ζ1611 | 2ζ165ζ32+ζ165-2ζ163ζ32-ζ163 | complex faithful |
►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 50 9 58)(2 49 10 57)(3 64 11 56)(4 63 12 55)(5 62 13 54)(6 61 14 53)(7 60 15 52)(8 59 16 51)(17 79 25 71)(18 78 26 70)(19 77 27 69)(20 76 28 68)(21 75 29 67)(22 74 30 66)(23 73 31 65)(24 72 32 80)(33 93 41 85)(34 92 42 84)(35 91 43 83)(36 90 44 82)(37 89 45 81)(38 88 46 96)(39 87 47 95)(40 86 48 94)
(1 69 88)(2 70 89)(3 71 90)(4 72 91)(5 73 92)(6 74 93)(7 75 94)(8 76 95)(9 77 96)(10 78 81)(11 79 82)(12 80 83)(13 65 84)(14 66 85)(15 67 86)(16 68 87)(17 44 64)(18 45 49)(19 46 50)(20 47 51)(21 48 52)(22 33 53)(23 34 54)(24 35 55)(25 36 56)(26 37 57)(27 38 58)(28 39 59)(29 40 60)(30 41 61)(31 42 62)(32 43 63)
(1 58)(2 51)(3 60)(4 53)(5 62)(6 55)(7 64)(8 57)(9 50)(10 59)(11 52)(12 61)(13 54)(14 63)(15 56)(16 49)(17 94)(18 87)(19 96)(20 89)(21 82)(22 91)(23 84)(24 93)(25 86)(26 95)(27 88)(28 81)(29 90)(30 83)(31 92)(32 85)(33 72)(34 65)(35 74)(36 67)(37 76)(38 69)(39 78)(40 71)(41 80)(42 73)(43 66)(44 75)(45 68)(46 77)(47 70)(48 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), (1,50,9,58)(2,49,10,57)(3,64,11,56)(4,63,12,55)(5,62,13,54)(6,61,14,53)(7,60,15,52)(8,59,16,51)(17,79,25,71)(18,78,26,70)(19,77,27,69)(20,76,28,68)(21,75,29,67)(22,74,30,66)(23,73,31,65)(24,72,32,80)(33,93,41,85)(34,92,42,84)(35,91,43,83)(36,90,44,82)(37,89,45,81)(38,88,46,96)(39,87,47,95)(40,86,48,94), (1,69,88)(2,70,89)(3,71,90)(4,72,91)(5,73,92)(6,74,93)(7,75,94)(8,76,95)(9,77,96)(10,78,81)(11,79,82)(12,80,83)(13,65,84)(14,66,85)(15,67,86)(16,68,87)(17,44,64)(18,45,49)(19,46,50)(20,47,51)(21,48,52)(22,33,53)(23,34,54)(24,35,55)(25,36,56)(26,37,57)(27,38,58)(28,39,59)(29,40,60)(30,41,61)(31,42,62)(32,43,63), (1,58)(2,51)(3,60)(4,53)(5,62)(6,55)(7,64)(8,57)(9,50)(10,59)(11,52)(12,61)(13,54)(14,63)(15,56)(16,49)(17,94)(18,87)(19,96)(20,89)(21,82)(22,91)(23,84)(24,93)(25,86)(26,95)(27,88)(28,81)(29,90)(30,83)(31,92)(32,85)(33,72)(34,65)(35,74)(36,67)(37,76)(38,69)(39,78)(40,71)(41,80)(42,73)(43,66)(44,75)(45,68)(46,77)(47,70)(48,79)>;
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), (1,50,9,58)(2,49,10,57)(3,64,11,56)(4,63,12,55)(5,62,13,54)(6,61,14,53)(7,60,15,52)(8,59,16,51)(17,79,25,71)(18,78,26,70)(19,77,27,69)(20,76,28,68)(21,75,29,67)(22,74,30,66)(23,73,31,65)(24,72,32,80)(33,93,41,85)(34,92,42,84)(35,91,43,83)(36,90,44,82)(37,89,45,81)(38,88,46,96)(39,87,47,95)(40,86,48,94), (1,69,88)(2,70,89)(3,71,90)(4,72,91)(5,73,92)(6,74,93)(7,75,94)(8,76,95)(9,77,96)(10,78,81)(11,79,82)(12,80,83)(13,65,84)(14,66,85)(15,67,86)(16,68,87)(17,44,64)(18,45,49)(19,46,50)(20,47,51)(21,48,52)(22,33,53)(23,34,54)(24,35,55)(25,36,56)(26,37,57)(27,38,58)(28,39,59)(29,40,60)(30,41,61)(31,42,62)(32,43,63), (1,58)(2,51)(3,60)(4,53)(5,62)(6,55)(7,64)(8,57)(9,50)(10,59)(11,52)(12,61)(13,54)(14,63)(15,56)(16,49)(17,94)(18,87)(19,96)(20,89)(21,82)(22,91)(23,84)(24,93)(25,86)(26,95)(27,88)(28,81)(29,90)(30,83)(31,92)(32,85)(33,72)(34,65)(35,74)(36,67)(37,76)(38,69)(39,78)(40,71)(41,80)(42,73)(43,66)(44,75)(45,68)(46,77)(47,70)(48,79) );
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)], [(1,50,9,58),(2,49,10,57),(3,64,11,56),(4,63,12,55),(5,62,13,54),(6,61,14,53),(7,60,15,52),(8,59,16,51),(17,79,25,71),(18,78,26,70),(19,77,27,69),(20,76,28,68),(21,75,29,67),(22,74,30,66),(23,73,31,65),(24,72,32,80),(33,93,41,85),(34,92,42,84),(35,91,43,83),(36,90,44,82),(37,89,45,81),(38,88,46,96),(39,87,47,95),(40,86,48,94)], [(1,69,88),(2,70,89),(3,71,90),(4,72,91),(5,73,92),(6,74,93),(7,75,94),(8,76,95),(9,77,96),(10,78,81),(11,79,82),(12,80,83),(13,65,84),(14,66,85),(15,67,86),(16,68,87),(17,44,64),(18,45,49),(19,46,50),(20,47,51),(21,48,52),(22,33,53),(23,34,54),(24,35,55),(25,36,56),(26,37,57),(27,38,58),(28,39,59),(29,40,60),(30,41,61),(31,42,62),(32,43,63)], [(1,58),(2,51),(3,60),(4,53),(5,62),(6,55),(7,64),(8,57),(9,50),(10,59),(11,52),(12,61),(13,54),(14,63),(15,56),(16,49),(17,94),(18,87),(19,96),(20,89),(21,82),(22,91),(23,84),(24,93),(25,86),(26,95),(27,88),(28,81),(29,90),(30,83),(31,92),(32,85),(33,72),(34,65),(35,74),(36,67),(37,76),(38,69),(39,78),(40,71),(41,80),(42,73),(43,66),(44,75),(45,68),(46,77),(47,70),(48,79)]])
Matrix representation of Q32⋊S3 ►in GL6(𝔽97)
| 96 | 0 | 0 | 0 | 0 | 0 |
| 0 | 96 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 51 | 14 | 12 |
| 0 | 0 | 16 | 66 | 1 | 13 |
| 0 | 0 | 82 | 68 | 15 | 29 |
| 0 | 0 | 15 | 67 | 81 | 14 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 95 | 0 |
| 0 | 0 | 0 | 0 | 96 | 1 |
| 0 | 0 | 1 | 0 | 96 | 0 |
| 0 | 0 | 1 | 96 | 96 | 0 |
| 0 | 96 | 0 | 0 | 0 | 0 |
| 1 | 96 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 95 | 0 |
| 0 | 0 | 1 | 0 | 96 | 96 |
| 0 | 0 | 0 | 0 | 96 | 0 |
| 0 | 0 | 1 | 96 | 96 | 0 |
G:=sub<GL(6,GF(97))| [96,0,0,0,0,0,0,96,0,0,0,0,0,0,2,16,82,15,0,0,51,66,68,67,0,0,14,1,15,81,0,0,12,13,29,14],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,96,0,0,95,96,96,96,0,0,0,1,0,0],[0,1,0,0,0,0,96,96,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,1,0,0,0,0,0,96,0,0,95,96,96,96,0,0,0,96,0,0] >;
Q32⋊S3 in GAP, Magma, Sage, TeX
Q_{32}\rtimes S_3 % in TeX
G:=Group("Q32:S3"); // GroupNames label
G:=SmallGroup(192,477);
// by ID
G=gap.SmallGroup(192,477);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,135,184,346,185,192,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^16=c^3=d^2=1,b^2=a^8,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^9,b*c=c*b,d*b*d=a^8*b,d*c*d=c^-1>;
// generators/relations
Export