p-group, metabelian, nilpotent (class 3), monomial
Aliases: D4.3SD16, C42.230C23, C8⋊2C8⋊5C2, C4⋊C4.205D4, (C8×D4).14C2, (C2×C8).313D4, (C2×D4).193D4, C4.70(C4○D8), C4.10D8⋊6C2, D4⋊Q8.5C2, C4.42(C2×SD16), C4⋊Q8.53C22, C2.14(C8⋊8D4), C4⋊C8.180C22, C4.90(C8⋊C22), (C4×C8).255C22, C4.SD16⋊23C2, D4.D4.8C2, C2.9(D4.2D4), (C4×D4).281C22, C2.10(D4.5D4), C22.191(C4⋊D4), (C2×C4).15(C4○D4), (C2×C4).1265(C2×D4), SmallGroup(128,411)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4.3SD16
G = < a,b,c,d | a4=b2=c8=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2c3 >
Subgroups: 184 in 83 conjugacy classes, 34 normal (32 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C2.D8, C4×D4, C4⋊Q8, C22×C8, C2×SD16, C4.10D8, C8⋊2C8, C8×D4, D4.D4, D4⋊Q8, C4.SD16, D4.3SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C4○D4, C4⋊D4, C2×SD16, C4○D8, C8⋊C22, D4.2D4, C8⋊8D4, D4.5D4, D4.3SD16
Character table of D4.3SD16
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 8M | 8N | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 16 | 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 | -2 | -2 | -2 | 2 | -2 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | 2 | -2 | 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 | 0 | 2 | -2 | 2 | -2 | 0 | -2 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | -2 | 0 | 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 | 0 | -2 | -2 | -2 | -2 | 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 | 0 | -2 | -2 | -2 | -2 | 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 | 0 | -2 | 0 | 2 | -2i | 0 | 2i | 0 | 0 | -√-2 | √-2 | √-2 | -√-2 | -√2 | √2 | -√2 | -√-2 | √-2 | √2 | 0 | 0 | 0 | 0 | complex lifted from C4○D8 |
| ρ16 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -2 | 0 | 2 | 2i | 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 | 2 | -2 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | -√-2 | √-2 | √-2 | -√-2 | √-2 | -√-2 | -√-2 | √-2 | -√-2 | √-2 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ18 | 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 |
| ρ19 | 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 |
| ρ20 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | -√-2 | √-2 | √-2 | -√-2 | -√-2 | √-2 | √-2 | √-2 | -√-2 | -√-2 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| 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 | 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 |
| ρ23 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | √-2 | -√-2 | -√-2 | √-2 | -√-2 | √-2 | √-2 | -√-2 | √-2 | -√-2 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ24 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -2 | 0 | 2 | -2i | 0 | 2i | 0 | 0 | √-2 | -√-2 | -√-2 | √-2 | √2 | -√2 | √2 | √-2 | -√-2 | -√2 | 0 | 0 | 0 | 0 | complex lifted from C4○D8 |
| ρ25 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -2 | 0 | 2 | 2i | 0 | -2i | 0 | 0 | -√-2 | √-2 | √-2 | -√-2 | √2 | -√2 | √2 | -√-2 | √-2 | -√2 | 0 | 0 | 0 | 0 | complex lifted from C4○D8 |
| ρ26 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 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 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| 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 | symplectic lifted from D4.5D4, Schur index 2 |
| ρ29 | 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 | symplectic lifted from D4.5D4, Schur index 2 |
►On 64 points
Generators in S64
(1 45 35 52)(2 46 36 53)(3 47 37 54)(4 48 38 55)(5 41 39 56)(6 42 40 49)(7 43 33 50)(8 44 34 51)(9 62 19 31)(10 63 20 32)(11 64 21 25)(12 57 22 26)(13 58 23 27)(14 59 24 28)(15 60 17 29)(16 61 18 30)
(1 56)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 60)(26 61)(27 62)(28 63)(29 64)(30 57)(31 58)(32 59)(33 47)(34 48)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)
(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)
(1 63 35 32)(2 27 36 58)(3 61 37 30)(4 25 38 64)(5 59 39 28)(6 31 40 62)(7 57 33 26)(8 29 34 60)(9 42 19 49)(10 52 20 45)(11 48 21 55)(12 50 22 43)(13 46 23 53)(14 56 24 41)(15 44 17 51)(16 54 18 47)
G:=sub<Sym(64)| (1,45,35,52)(2,46,36,53)(3,47,37,54)(4,48,38,55)(5,41,39,56)(6,42,40,49)(7,43,33,50)(8,44,34,51)(9,62,19,31)(10,63,20,32)(11,64,21,25)(12,57,22,26)(13,58,23,27)(14,59,24,28)(15,60,17,29)(16,61,18,30), (1,56)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,60)(26,61)(27,62)(28,63)(29,64)(30,57)(31,58)(32,59)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (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), (1,63,35,32)(2,27,36,58)(3,61,37,30)(4,25,38,64)(5,59,39,28)(6,31,40,62)(7,57,33,26)(8,29,34,60)(9,42,19,49)(10,52,20,45)(11,48,21,55)(12,50,22,43)(13,46,23,53)(14,56,24,41)(15,44,17,51)(16,54,18,47)>;
G:=Group( (1,45,35,52)(2,46,36,53)(3,47,37,54)(4,48,38,55)(5,41,39,56)(6,42,40,49)(7,43,33,50)(8,44,34,51)(9,62,19,31)(10,63,20,32)(11,64,21,25)(12,57,22,26)(13,58,23,27)(14,59,24,28)(15,60,17,29)(16,61,18,30), (1,56)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,60)(26,61)(27,62)(28,63)(29,64)(30,57)(31,58)(32,59)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (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), (1,63,35,32)(2,27,36,58)(3,61,37,30)(4,25,38,64)(5,59,39,28)(6,31,40,62)(7,57,33,26)(8,29,34,60)(9,42,19,49)(10,52,20,45)(11,48,21,55)(12,50,22,43)(13,46,23,53)(14,56,24,41)(15,44,17,51)(16,54,18,47) );
G=PermutationGroup([[(1,45,35,52),(2,46,36,53),(3,47,37,54),(4,48,38,55),(5,41,39,56),(6,42,40,49),(7,43,33,50),(8,44,34,51),(9,62,19,31),(10,63,20,32),(11,64,21,25),(12,57,22,26),(13,58,23,27),(14,59,24,28),(15,60,17,29),(16,61,18,30)], [(1,56),(2,49),(3,50),(4,51),(5,52),(6,53),(7,54),(8,55),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,60),(26,61),(27,62),(28,63),(29,64),(30,57),(31,58),(32,59),(33,47),(34,48),(35,41),(36,42),(37,43),(38,44),(39,45),(40,46)], [(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)], [(1,63,35,32),(2,27,36,58),(3,61,37,30),(4,25,38,64),(5,59,39,28),(6,31,40,62),(7,57,33,26),(8,29,34,60),(9,42,19,49),(10,52,20,45),(11,48,21,55),(12,50,22,43),(13,46,23,53),(14,56,24,41),(15,44,17,51),(16,54,18,47)]])
Matrix representation of D4.3SD16 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 5 | 5 | 0 | 0 |
| 12 | 5 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 5 | 12 |
| 0 | 0 | 12 | 12 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,0,16,0,0,1,0],[16,0,0,0,0,16,0,0,0,0,0,1,0,0,1,0],[5,12,0,0,5,5,0,0,0,0,13,0,0,0,0,13],[16,0,0,0,0,1,0,0,0,0,5,12,0,0,12,12] >;
D4.3SD16 in GAP, Magma, Sage, TeX
D_4._3{\rm SD}_{16} % in TeX
G:=Group("D4.3SD16"); // GroupNames label
G:=SmallGroup(128,411);
// by ID
G=gap.SmallGroup(128,411);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,512,422,1123,136,2804,718,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^8=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=a^2*c^3>;
// generators/relations
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