p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q8.2SD16, C42.229C23, C8⋊2C8⋊4C2, (C8×Q8)⋊20C2, C4⋊C4.204D4, C4.Q16⋊6C2, (C2×C8).312D4, C4.69(C4○D8), C4⋊C8.24C22, (C2×Q8).152D4, C4⋊SD16.7C2, C4.41(C2×SD16), C4.D8.2C2, C4⋊Q8.52C22, C2.13(C8⋊8D4), (C4×C8).254C22, C4.6Q16⋊16C2, C4.4D8.11C2, C4⋊1D4.29C22, C4.91(C8.C22), C2.8(Q8.D4), (C4×Q8).267C22, C2.11(D4.4D4), C22.190(C4⋊D4), (C2×C4).14(C4○D4), (C2×C4).1264(C2×D4), SmallGroup(128,410)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q8.2SD16
G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, dbd=a-1b, dcd=a2c3 >
Subgroups: 200 in 80 conjugacy classes, 34 normal (32 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C2×D4, C2×Q8, C2×Q8, C4×C8, C4×C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C2.D8, C4×Q8, C4⋊1D4, C4⋊Q8, C2×SD16, C4.D8, C4.6Q16, C8⋊2C8, C8×Q8, C4⋊SD16, C4.Q16, C4.4D8, Q8.2SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C4○D4, C4⋊D4, C2×SD16, C4○D8, C8.C22, Q8.D4, C8⋊8D4, D4.4D4, Q8.2SD16
Character table of Q8.2SD16
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 8M | 8N | |
| size | 1 | 1 | 1 | 1 | 16 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 16 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | |
| 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | -2i | 2i | 0 | 0 | 2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 2i | -2i | 0 | 0 | -2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ15 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | -√-2 | √2 | √-2 | complex lifted from C4○D8 |
| ρ16 | 2 | 2 | -2 | -2 | 0 | -2 | 0 | 2 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | -√-2 | √-2 | -√-2 | √-2 | -√2 | √2 | -√2 | -√-2 | √-2 | √2 | 0 | 0 | 0 | 0 | complex lifted from C4○D8 |
| ρ17 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√-2 | -√2 | √-2 | complex lifted from C4○D8 |
| ρ18 | 2 | 2 | -2 | -2 | 0 | 2 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | -√-2 | √-2 | -√-2 | √-2 | √-2 | -√-2 | -√-2 | √-2 | -√-2 | √-2 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ19 | 2 | 2 | -2 | -2 | 0 | 2 | 0 | -2 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | √-2 | -√-2 | √-2 | -√-2 | √-2 | -√-2 | -√-2 | -√-2 | √-2 | √-2 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ20 | 2 | 2 | -2 | -2 | 0 | -2 | 0 | 2 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | √-2 | -√-2 | √-2 | -√-2 | -√2 | √2 | -√2 | √-2 | -√-2 | √2 | 0 | 0 | 0 | 0 | complex lifted from C4○D8 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | √2 | √-2 | -√2 | -√-2 | complex lifted from C4○D8 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | -2 | 0 | 2 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | -√-2 | √-2 | -√-2 | √-2 | √2 | -√2 | √2 | -√-2 | √-2 | -√2 | 0 | 0 | 0 | 0 | complex lifted from C4○D8 |
| ρ23 | 2 | 2 | -2 | -2 | 0 | -2 | 0 | 2 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | √-2 | -√-2 | √-2 | -√-2 | √2 | -√2 | √2 | √-2 | -√-2 | -√2 | 0 | 0 | 0 | 0 | complex lifted from C4○D8 |
| ρ24 | 2 | 2 | -2 | -2 | 0 | 2 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | √-2 | -√-2 | √-2 | -√-2 | -√-2 | √-2 | √-2 | -√-2 | √-2 | -√-2 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ25 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √-2 | √2 | -√-2 | complex lifted from C4○D8 |
| ρ26 | 2 | 2 | -2 | -2 | 0 | 2 | 0 | -2 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | -√-2 | √-2 | -√-2 | √-2 | -√-2 | √-2 | √-2 | √-2 | -√-2 | -√-2 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ27 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | 2√2 | -2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4.4D4 |
| ρ28 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | -2√2 | 2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4.4D4 |
| ρ29 | 4 | -4 | 4 | -4 | 0 | 0 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
►On 64 points
Generators in S64
(1 61 52 11)(2 62 53 12)(3 63 54 13)(4 64 55 14)(5 57 56 15)(6 58 49 16)(7 59 50 9)(8 60 51 10)(17 36 25 47)(18 37 26 48)(19 38 27 41)(20 39 28 42)(21 40 29 43)(22 33 30 44)(23 34 31 45)(24 35 32 46)
(1 17 52 25)(2 18 53 26)(3 19 54 27)(4 20 55 28)(5 21 56 29)(6 22 49 30)(7 23 50 31)(8 24 51 32)(9 34 59 45)(10 35 60 46)(11 36 61 47)(12 37 62 48)(13 38 63 41)(14 39 64 42)(15 40 57 43)(16 33 58 44)
(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)
(2 55)(3 7)(4 53)(6 51)(8 49)(9 63)(10 16)(11 61)(12 14)(13 59)(15 57)(17 36)(18 42)(19 34)(20 48)(21 40)(22 46)(23 38)(24 44)(25 47)(26 39)(27 45)(28 37)(29 43)(30 35)(31 41)(32 33)(50 54)(58 60)(62 64)
G:=sub<Sym(64)| (1,61,52,11)(2,62,53,12)(3,63,54,13)(4,64,55,14)(5,57,56,15)(6,58,49,16)(7,59,50,9)(8,60,51,10)(17,36,25,47)(18,37,26,48)(19,38,27,41)(20,39,28,42)(21,40,29,43)(22,33,30,44)(23,34,31,45)(24,35,32,46), (1,17,52,25)(2,18,53,26)(3,19,54,27)(4,20,55,28)(5,21,56,29)(6,22,49,30)(7,23,50,31)(8,24,51,32)(9,34,59,45)(10,35,60,46)(11,36,61,47)(12,37,62,48)(13,38,63,41)(14,39,64,42)(15,40,57,43)(16,33,58,44), (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), (2,55)(3,7)(4,53)(6,51)(8,49)(9,63)(10,16)(11,61)(12,14)(13,59)(15,57)(17,36)(18,42)(19,34)(20,48)(21,40)(22,46)(23,38)(24,44)(25,47)(26,39)(27,45)(28,37)(29,43)(30,35)(31,41)(32,33)(50,54)(58,60)(62,64)>;
G:=Group( (1,61,52,11)(2,62,53,12)(3,63,54,13)(4,64,55,14)(5,57,56,15)(6,58,49,16)(7,59,50,9)(8,60,51,10)(17,36,25,47)(18,37,26,48)(19,38,27,41)(20,39,28,42)(21,40,29,43)(22,33,30,44)(23,34,31,45)(24,35,32,46), (1,17,52,25)(2,18,53,26)(3,19,54,27)(4,20,55,28)(5,21,56,29)(6,22,49,30)(7,23,50,31)(8,24,51,32)(9,34,59,45)(10,35,60,46)(11,36,61,47)(12,37,62,48)(13,38,63,41)(14,39,64,42)(15,40,57,43)(16,33,58,44), (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), (2,55)(3,7)(4,53)(6,51)(8,49)(9,63)(10,16)(11,61)(12,14)(13,59)(15,57)(17,36)(18,42)(19,34)(20,48)(21,40)(22,46)(23,38)(24,44)(25,47)(26,39)(27,45)(28,37)(29,43)(30,35)(31,41)(32,33)(50,54)(58,60)(62,64) );
G=PermutationGroup([[(1,61,52,11),(2,62,53,12),(3,63,54,13),(4,64,55,14),(5,57,56,15),(6,58,49,16),(7,59,50,9),(8,60,51,10),(17,36,25,47),(18,37,26,48),(19,38,27,41),(20,39,28,42),(21,40,29,43),(22,33,30,44),(23,34,31,45),(24,35,32,46)], [(1,17,52,25),(2,18,53,26),(3,19,54,27),(4,20,55,28),(5,21,56,29),(6,22,49,30),(7,23,50,31),(8,24,51,32),(9,34,59,45),(10,35,60,46),(11,36,61,47),(12,37,62,48),(13,38,63,41),(14,39,64,42),(15,40,57,43),(16,33,58,44)], [(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)], [(2,55),(3,7),(4,53),(6,51),(8,49),(9,63),(10,16),(11,61),(12,14),(13,59),(15,57),(17,36),(18,42),(19,34),(20,48),(21,40),(22,46),(23,38),(24,44),(25,47),(26,39),(27,45),(28,37),(29,43),(30,35),(31,41),(32,33),(50,54),(58,60),(62,64)]])
Matrix representation of Q8.2SD16 ►in GL4(𝔽17) generated by
| 16 | 15 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 10 | 0 | 0 |
| 5 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 10 | 10 |
| 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 16 | 16 |
G:=sub<GL(4,GF(17))| [16,1,0,0,15,1,0,0,0,0,1,0,0,0,0,1],[0,5,0,0,10,0,0,0,0,0,16,0,0,0,0,16],[13,0,0,0,0,13,0,0,0,0,10,12,0,0,10,0],[1,16,0,0,0,16,0,0,0,0,1,16,0,0,0,16] >;
Q8.2SD16 in GAP, Magma, Sage, TeX
Q_8._2{\rm SD}_{16} % in TeX
G:=Group("Q8.2SD16"); // GroupNames label
G:=SmallGroup(128,410);
// by ID
G=gap.SmallGroup(128,410);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,512,422,352,1123,136,2804,718,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^8=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a^-1*b,d*c*d=a^2*c^3>;
// generators/relations
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