metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8.4D6, D6.8D8, C16.2D6, SD32:2S3, Q16.1D6, Dic24:6C2, C48.9C22, C24.18C23, Dic3.10D8, Dic12.3C22, C3:C8.4D4, C4.6(S3xD4), D6.C8:2C2, D8:3S3.C2, D8.S3:4C2, (S3xQ16):4C2, C3:Q32:1C2, (C4xS3).9D4, C6.37(C2xD8), C2.21(S3xD8), (C3xSD32):2C2, C12.12(C2xD4), C3:2(Q32:C2), C3:C16.1C22, (S3xC8).5C22, C8.24(C22xS3), (C3xD8).4C22, (C3xQ16).2C22, SmallGroup(192,474)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for SD32:S3
G = < a,b,c,d | a16=b2=c3=d2=1, bab=a7, ac=ca, dad=a9, bc=cb, dbd=a8b, dcd=c-1 >
Subgroups: 268 in 82 conjugacy classes, 31 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2xC4, D4, Q8, Dic3, Dic3, C12, C12, D6, C2xC6, C16, C16, C2xC8, D8, SD16, Q16, Q16, C2xQ8, C4oD4, C3:C8, C24, Dic6, C4xS3, C4xS3, C2xDic3, C3:D4, C3xD4, C3xQ8, M5(2), SD32, SD32, Q32, C2xQ16, C4oD8, C3:C16, C48, S3xC8, Dic12, D4.S3, C3:Q16, C3xD8, C3xQ16, D4:2S3, S3xQ8, Q32:C2, D6.C8, Dic24, D8.S3, C3:Q32, C3xSD32, D8:3S3, S3xQ16, SD32:S3
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2xD4, C22xS3, C2xD8, S3xD4, Q32:C2, S3xD8, SD32:S3
Character table of SD32:S3
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 8A | 8B | 8C | 12A | 12B | 16A | 16B | 16C | 16D | 24A | 24B | 48A | 48B | 48C | 48D | |
| size | 1 | 1 | 6 | 8 | 2 | 2 | 6 | 8 | 24 | 24 | 2 | 16 | 2 | 2 | 12 | 4 | 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 | -2 | -1 | 2 | 0 | 2 | 0 | 0 | -1 | 1 | 2 | 2 | 0 | -1 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | 0 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 0 | 2 | -1 | 2 | 0 | -2 | 0 | 0 | -1 | -1 | 2 | 2 | 0 | -1 | 1 | -2 | -2 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 0 | -2 | -1 | 2 | 0 | -2 | 0 | 0 | -1 | 1 | 2 | 2 | 0 | -1 | 1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | 0 | 2 | -1 | 2 | 0 | 2 | 0 | 0 | -1 | -1 | 2 | 2 | 0 | -1 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ14 | 2 | 2 | -2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | -2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 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 | 0 | -2 | 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 | 0 | -2 | 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 | 0 | -2 | 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 | 0 | -4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from S3xD4 |
| ρ20 | 4 | 4 | 0 | 0 | -2 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | √2 | √2 | -√2 | -√2 | orthogonal lifted from S3xD8 |
| ρ21 | 4 | 4 | 0 | 0 | -2 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | -√2 | -√2 | √2 | √2 | orthogonal lifted from S3xD8 |
| ρ22 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | -2√2 | 2√2 | 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 | 0 | 2√2 | -2√2 | 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 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | 2ζ165ζ3+ζ165+2ζ163ζ3+ζ163 | 2ζ165ζ32+ζ165+2ζ163ζ32+ζ163 | 2ζ1615ζ32+ζ1615+2ζ169ζ32+ζ169 | 2ζ167ζ32+ζ167+2ζ16ζ32+ζ16 | symplectic faithful, Schur index 2 |
| ρ25 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | 2ζ167ζ32+ζ167+2ζ16ζ32+ζ16 | 2ζ1615ζ32+ζ1615+2ζ169ζ32+ζ169 | 2ζ165ζ3+ζ165+2ζ163ζ3+ζ163 | 2ζ165ζ32+ζ165+2ζ163ζ32+ζ163 | symplectic faithful, Schur index 2 |
| ρ26 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | 2ζ165ζ32+ζ165+2ζ163ζ32+ζ163 | 2ζ165ζ3+ζ165+2ζ163ζ3+ζ163 | 2ζ167ζ32+ζ167+2ζ16ζ32+ζ16 | 2ζ1615ζ32+ζ1615+2ζ169ζ32+ζ169 | symplectic faithful, Schur index 2 |
| ρ27 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | 2ζ1615ζ32+ζ1615+2ζ169ζ32+ζ169 | 2ζ167ζ32+ζ167+2ζ16ζ32+ζ16 | 2ζ165ζ32+ζ165+2ζ163ζ32+ζ163 | 2ζ165ζ3+ζ165+2ζ163ζ3+ζ163 | 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)
(2 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 25)(18 32)(19 23)(20 30)(22 28)(24 26)(27 31)(33 45)(34 36)(35 43)(37 41)(38 48)(40 46)(42 44)(49 57)(50 64)(51 55)(52 62)(54 60)(56 58)(59 63)(65 69)(66 76)(68 74)(70 72)(71 79)(73 77)(78 80)(81 83)(82 90)(84 88)(85 95)(87 93)(89 91)(92 96)
(1 94 39)(2 95 40)(3 96 41)(4 81 42)(5 82 43)(6 83 44)(7 84 45)(8 85 46)(9 86 47)(10 87 48)(11 88 33)(12 89 34)(13 90 35)(14 91 36)(15 92 37)(16 93 38)(17 57 79)(18 58 80)(19 59 65)(20 60 66)(21 61 67)(22 62 68)(23 63 69)(24 64 70)(25 49 71)(26 50 72)(27 51 73)(28 52 74)(29 53 75)(30 54 76)(31 55 77)(32 56 78)
(1 25)(2 18)(3 27)(4 20)(5 29)(6 22)(7 31)(8 24)(9 17)(10 26)(11 19)(12 28)(13 21)(14 30)(15 23)(16 32)(33 59)(34 52)(35 61)(36 54)(37 63)(38 56)(39 49)(40 58)(41 51)(42 60)(43 53)(44 62)(45 55)(46 64)(47 57)(48 50)(65 88)(66 81)(67 90)(68 83)(69 92)(70 85)(71 94)(72 87)(73 96)(74 89)(75 82)(76 91)(77 84)(78 93)(79 86)(80 95)
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), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,25)(18,32)(19,23)(20,30)(22,28)(24,26)(27,31)(33,45)(34,36)(35,43)(37,41)(38,48)(40,46)(42,44)(49,57)(50,64)(51,55)(52,62)(54,60)(56,58)(59,63)(65,69)(66,76)(68,74)(70,72)(71,79)(73,77)(78,80)(81,83)(82,90)(84,88)(85,95)(87,93)(89,91)(92,96), (1,94,39)(2,95,40)(3,96,41)(4,81,42)(5,82,43)(6,83,44)(7,84,45)(8,85,46)(9,86,47)(10,87,48)(11,88,33)(12,89,34)(13,90,35)(14,91,36)(15,92,37)(16,93,38)(17,57,79)(18,58,80)(19,59,65)(20,60,66)(21,61,67)(22,62,68)(23,63,69)(24,64,70)(25,49,71)(26,50,72)(27,51,73)(28,52,74)(29,53,75)(30,54,76)(31,55,77)(32,56,78), (1,25)(2,18)(3,27)(4,20)(5,29)(6,22)(7,31)(8,24)(9,17)(10,26)(11,19)(12,28)(13,21)(14,30)(15,23)(16,32)(33,59)(34,52)(35,61)(36,54)(37,63)(38,56)(39,49)(40,58)(41,51)(42,60)(43,53)(44,62)(45,55)(46,64)(47,57)(48,50)(65,88)(66,81)(67,90)(68,83)(69,92)(70,85)(71,94)(72,87)(73,96)(74,89)(75,82)(76,91)(77,84)(78,93)(79,86)(80,95)>;
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), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,25)(18,32)(19,23)(20,30)(22,28)(24,26)(27,31)(33,45)(34,36)(35,43)(37,41)(38,48)(40,46)(42,44)(49,57)(50,64)(51,55)(52,62)(54,60)(56,58)(59,63)(65,69)(66,76)(68,74)(70,72)(71,79)(73,77)(78,80)(81,83)(82,90)(84,88)(85,95)(87,93)(89,91)(92,96), (1,94,39)(2,95,40)(3,96,41)(4,81,42)(5,82,43)(6,83,44)(7,84,45)(8,85,46)(9,86,47)(10,87,48)(11,88,33)(12,89,34)(13,90,35)(14,91,36)(15,92,37)(16,93,38)(17,57,79)(18,58,80)(19,59,65)(20,60,66)(21,61,67)(22,62,68)(23,63,69)(24,64,70)(25,49,71)(26,50,72)(27,51,73)(28,52,74)(29,53,75)(30,54,76)(31,55,77)(32,56,78), (1,25)(2,18)(3,27)(4,20)(5,29)(6,22)(7,31)(8,24)(9,17)(10,26)(11,19)(12,28)(13,21)(14,30)(15,23)(16,32)(33,59)(34,52)(35,61)(36,54)(37,63)(38,56)(39,49)(40,58)(41,51)(42,60)(43,53)(44,62)(45,55)(46,64)(47,57)(48,50)(65,88)(66,81)(67,90)(68,83)(69,92)(70,85)(71,94)(72,87)(73,96)(74,89)(75,82)(76,91)(77,84)(78,93)(79,86)(80,95) );
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)], [(2,8),(3,15),(4,6),(5,13),(7,11),(10,16),(12,14),(17,25),(18,32),(19,23),(20,30),(22,28),(24,26),(27,31),(33,45),(34,36),(35,43),(37,41),(38,48),(40,46),(42,44),(49,57),(50,64),(51,55),(52,62),(54,60),(56,58),(59,63),(65,69),(66,76),(68,74),(70,72),(71,79),(73,77),(78,80),(81,83),(82,90),(84,88),(85,95),(87,93),(89,91),(92,96)], [(1,94,39),(2,95,40),(3,96,41),(4,81,42),(5,82,43),(6,83,44),(7,84,45),(8,85,46),(9,86,47),(10,87,48),(11,88,33),(12,89,34),(13,90,35),(14,91,36),(15,92,37),(16,93,38),(17,57,79),(18,58,80),(19,59,65),(20,60,66),(21,61,67),(22,62,68),(23,63,69),(24,64,70),(25,49,71),(26,50,72),(27,51,73),(28,52,74),(29,53,75),(30,54,76),(31,55,77),(32,56,78)], [(1,25),(2,18),(3,27),(4,20),(5,29),(6,22),(7,31),(8,24),(9,17),(10,26),(11,19),(12,28),(13,21),(14,30),(15,23),(16,32),(33,59),(34,52),(35,61),(36,54),(37,63),(38,56),(39,49),(40,58),(41,51),(42,60),(43,53),(44,62),(45,55),(46,64),(47,57),(48,50),(65,88),(66,81),(67,90),(68,83),(69,92),(70,85),(71,94),(72,87),(73,96),(74,89),(75,82),(76,91),(77,84),(78,93),(79,86),(80,95)]])
Matrix representation of SD32:S3 ►in GL4(F7) generated by
| 0 | 5 | 3 | 6 |
| 3 | 2 | 3 | 3 |
| 6 | 2 | 2 | 6 |
| 5 | 4 | 5 | 3 |
| 5 | 3 | 1 | 6 |
| 1 | 0 | 0 | 5 |
| 0 | 0 | 6 | 0 |
| 6 | 1 | 4 | 3 |
| 0 | 0 | 5 | 2 |
| 5 | 3 | 0 | 3 |
| 2 | 5 | 0 | 3 |
| 5 | 5 | 1 | 2 |
| 3 | 2 | 2 | 2 |
| 3 | 0 | 1 | 0 |
| 5 | 2 | 1 | 1 |
| 2 | 2 | 2 | 3 |
G:=sub<GL(4,GF(7))| [0,3,6,5,5,2,2,4,3,3,2,5,6,3,6,3],[5,1,0,6,3,0,0,1,1,0,6,4,6,5,0,3],[0,5,2,5,0,3,5,5,5,0,0,1,2,3,3,2],[3,3,5,2,2,0,2,2,2,1,1,2,2,0,1,3] >;
SD32:S3 in GAP, Magma, Sage, TeX
{\rm SD}_{32}\rtimes S_3 % in TeX
G:=Group("SD32:S3"); // GroupNames label
G:=SmallGroup(192,474);
// by ID
G=gap.SmallGroup(192,474);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,135,346,185,192,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^16=b^2=c^3=d^2=1,b*a*b=a^7,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
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