p-group, metabelian, nilpotent (class 3), monomial
Aliases: D4.2SD16, C42.228C23, C8⋊2C8⋊3C2, (C8×D4)⋊29C2, C4⋊C4.203D4, C4⋊D8.5C2, (C2×C8).311D4, D4⋊2Q8⋊33C2, (C2×D4).192D4, C4.68(C4○D8), C4.D8⋊15C2, C4.4D8⋊24C2, C4.6Q16⋊4C2, C4.40(C2×SD16), C4⋊Q8.51C22, C2.12(C8⋊8D4), C4⋊C8.179C22, C4.89(C8⋊C22), (C4×C8).253C22, C2.8(D4.2D4), (C4×D4).280C22, C4⋊1D4.28C22, C2.10(D4.4D4), C22.189(C4⋊D4), (C2×C4).13(C4○D4), (C2×C4).1263(C2×D4), SmallGroup(128,409)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4.2SD16
G = < a,b,c,d | a4=b2=c8=d2=1, bab=dad=a-1, ac=ca, bc=cb, dbd=ab, dcd=a2c3 >
Subgroups: 232 in 88 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, D8, C22×C4, C2×D4, C2×D4, C2×Q8, C4×C8, C22⋊C8, D4⋊C4, C4⋊C8, C4.Q8, C4×D4, C4⋊1D4, C4⋊Q8, C22×C8, C2×D8, C4.D8, C4.6Q16, C8⋊2C8, C8×D4, C4⋊D8, D4⋊2Q8, C4.4D8, D4.2SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C4○D4, C4⋊D4, C2×SD16, C4○D8, C8⋊C22, D4.2D4, C8⋊8D4, D4.4D4, D4.2SD16
Character table of D4.2SD16
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 8M | 8N | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 16 | 2 | 2 | 2 | 2 | 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 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | -2 | 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 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | -2 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 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 | 2 | 2 | 0 | 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 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 2 | 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 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 2 | 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 | 0 | 2i | -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 | -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 |
| ρ17 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | -2i | 2i | 0 | 0 | -√-2 | √-2 | -√-2 | √-2 | √2 | -√2 | √2 | -√-2 | √-2 | -√2 | 0 | 0 | 0 | 0 | complex lifted from C4○D8 |
| ρ18 | 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 |
| ρ19 | 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 |
| ρ20 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | -2i | 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 | 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 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 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 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | √2 | √-2 | -√2 | complex lifted from C4○D8 |
| ρ24 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 2i | -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 | 0 | 2 | 0 | -2 | 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 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | -√2 | √-2 | √2 | complex lifted from C4○D8 |
| ρ27 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | -4 | 0 | 4 | 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 | orthogonal lifted from D4.4D4 |
| 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 | orthogonal lifted from D4.4D4 |
►On 64 points
Generators in S64
(1 61 22 34)(2 62 23 35)(3 63 24 36)(4 64 17 37)(5 57 18 38)(6 58 19 39)(7 59 20 40)(8 60 21 33)(9 49 43 31)(10 50 44 32)(11 51 45 25)(12 52 46 26)(13 53 47 27)(14 54 48 28)(15 55 41 29)(16 56 42 30)
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 41)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 57)(32 58)(33 52)(34 53)(35 54)(36 55)(37 56)(38 49)(39 50)(40 51)
(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 17)(3 7)(4 23)(6 21)(8 19)(9 31)(10 52)(11 29)(12 50)(13 27)(14 56)(15 25)(16 54)(20 24)(26 44)(28 42)(30 48)(32 46)(33 39)(34 61)(35 37)(36 59)(38 57)(40 63)(41 51)(43 49)(45 55)(47 53)(58 60)(62 64)
G:=sub<Sym(64)| (1,61,22,34)(2,62,23,35)(3,63,24,36)(4,64,17,37)(5,57,18,38)(6,58,19,39)(7,59,20,40)(8,60,21,33)(9,49,43,31)(10,50,44,32)(11,51,45,25)(12,52,46,26)(13,53,47,27)(14,54,48,28)(15,55,41,29)(16,56,42,30), (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,41)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,57)(32,58)(33,52)(34,53)(35,54)(36,55)(37,56)(38,49)(39,50)(40,51), (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), (2,17)(3,7)(4,23)(6,21)(8,19)(9,31)(10,52)(11,29)(12,50)(13,27)(14,56)(15,25)(16,54)(20,24)(26,44)(28,42)(30,48)(32,46)(33,39)(34,61)(35,37)(36,59)(38,57)(40,63)(41,51)(43,49)(45,55)(47,53)(58,60)(62,64)>;
G:=Group( (1,61,22,34)(2,62,23,35)(3,63,24,36)(4,64,17,37)(5,57,18,38)(6,58,19,39)(7,59,20,40)(8,60,21,33)(9,49,43,31)(10,50,44,32)(11,51,45,25)(12,52,46,26)(13,53,47,27)(14,54,48,28)(15,55,41,29)(16,56,42,30), (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,41)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,57)(32,58)(33,52)(34,53)(35,54)(36,55)(37,56)(38,49)(39,50)(40,51), (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), (2,17)(3,7)(4,23)(6,21)(8,19)(9,31)(10,52)(11,29)(12,50)(13,27)(14,56)(15,25)(16,54)(20,24)(26,44)(28,42)(30,48)(32,46)(33,39)(34,61)(35,37)(36,59)(38,57)(40,63)(41,51)(43,49)(45,55)(47,53)(58,60)(62,64) );
G=PermutationGroup([[(1,61,22,34),(2,62,23,35),(3,63,24,36),(4,64,17,37),(5,57,18,38),(6,58,19,39),(7,59,20,40),(8,60,21,33),(9,49,43,31),(10,50,44,32),(11,51,45,25),(12,52,46,26),(13,53,47,27),(14,54,48,28),(15,55,41,29),(16,56,42,30)], [(1,13),(2,14),(3,15),(4,16),(5,9),(6,10),(7,11),(8,12),(17,42),(18,43),(19,44),(20,45),(21,46),(22,47),(23,48),(24,41),(25,59),(26,60),(27,61),(28,62),(29,63),(30,64),(31,57),(32,58),(33,52),(34,53),(35,54),(36,55),(37,56),(38,49),(39,50),(40,51)], [(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)], [(2,17),(3,7),(4,23),(6,21),(8,19),(9,31),(10,52),(11,29),(12,50),(13,27),(14,56),(15,25),(16,54),(20,24),(26,44),(28,42),(30,48),(32,46),(33,39),(34,61),(35,37),(36,59),(38,57),(40,63),(41,51),(43,49),(45,55),(47,53),(58,60),(62,64)]])
Matrix representation of D4.2SD16 ►in GL4(𝔽17) generated by
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 3 | 14 | 0 | 0 |
| 14 | 14 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 10 |
| 0 | 0 | 12 | 7 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 16 | 16 |
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[3,14,0,0,14,14,0,0,0,0,1,0,0,0,0,1],[13,0,0,0,0,13,0,0,0,0,0,12,0,0,10,7],[1,0,0,0,0,16,0,0,0,0,1,16,0,0,0,16] >;
D4.2SD16 in GAP, Magma, Sage, TeX
D_4._2{\rm SD}_{16} % in TeX
G:=Group("D4.2SD16"); // GroupNames label
G:=SmallGroup(128,409);
// by ID
G=gap.SmallGroup(128,409);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,512,422,1123,136,2804,718,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^8=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a*b,d*c*d=a^2*c^3>;
// generators/relations
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