p-group, metabelian, nilpotent (class 3), monomial
Aliases: D4.2D8, C42.232C23, (C8×D4)⋊6C2, C8⋊1C8⋊8C2, C4.35(C2×D8), C4⋊C4.207D4, (C2×C8).315D4, C4.4D8⋊9C2, C4⋊D8.6C2, (C2×D4).194D4, C4.88(C4○D8), C4.10D8⋊7C2, C4.D8⋊16C2, (C4×C8).57C22, C4⋊Q8.55C22, D4.D4⋊35C2, C2.10(C8⋊7D4), C4⋊C8.181C22, C4.91(C8⋊C22), (C4×D4).282C22, C4⋊1D4.30C22, C2.10(D4.2D4), C2.12(D4.3D4), C22.193(C4⋊D4), (C2×C4).17(C4○D4), (C2×C4).1267(C2×D4), SmallGroup(128,413)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4.2D8
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=a2c-1 >
Subgroups: 240 in 91 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, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4⋊1D4, C4⋊Q8, C22×C8, C2×D8, C2×SD16, C4.D8, C4.10D8, C8⋊1C8, C8×D4, C4⋊D8, D4.D4, C4.4D8, D4.2D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C4○D4, C4⋊D4, C2×D8, C4○D8, C8⋊C22, D4.2D4, C8⋊7D4, D4.3D4, D4.2D8
Character table of D4.2D8
| 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 | 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 | orthogonal lifted from D8 |
| ρ14 | 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 | orthogonal lifted from D8 |
| ρ15 | 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 | orthogonal lifted from 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 | orthogonal lifted from D8 |
| ρ17 | 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 |
| ρ18 | 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 |
| ρ19 | 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 |
| 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 | 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 |
| 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 | complex lifted from D4.3D4 |
| 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 | complex lifted from D4.3D4 |
►On 64 points
Generators in S64
(1 61 22 38)(2 62 23 39)(3 63 24 40)(4 64 17 33)(5 57 18 34)(6 58 19 35)(7 59 20 36)(8 60 21 37)(9 52 47 27)(10 53 48 28)(11 54 41 29)(12 55 42 30)(13 56 43 31)(14 49 44 32)(15 50 45 25)(16 51 46 26)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 44)(18 45)(19 46)(20 47)(21 48)(22 41)(23 42)(24 43)(25 57)(26 58)(27 59)(28 60)(29 61)(30 62)(31 63)(32 64)(33 49)(34 50)(35 51)(36 52)(37 53)(38 54)(39 55)(40 56)
(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 21 22 8)(2 7 23 20)(3 19 24 6)(4 5 17 18)(9 55 47 30)(10 29 48 54)(11 53 41 28)(12 27 42 52)(13 51 43 26)(14 25 44 50)(15 49 45 32)(16 31 46 56)(33 57 64 34)(35 63 58 40)(36 39 59 62)(37 61 60 38)
G:=sub<Sym(64)| (1,61,22,38)(2,62,23,39)(3,63,24,40)(4,64,17,33)(5,57,18,34)(6,58,19,35)(7,59,20,36)(8,60,21,37)(9,52,47,27)(10,53,48,28)(11,54,41,29)(12,55,42,30)(13,56,43,31)(14,49,44,32)(15,50,45,25)(16,51,46,26), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,44)(18,45)(19,46)(20,47)(21,48)(22,41)(23,42)(24,43)(25,57)(26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(32,64)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56), (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), (1,21,22,8)(2,7,23,20)(3,19,24,6)(4,5,17,18)(9,55,47,30)(10,29,48,54)(11,53,41,28)(12,27,42,52)(13,51,43,26)(14,25,44,50)(15,49,45,32)(16,31,46,56)(33,57,64,34)(35,63,58,40)(36,39,59,62)(37,61,60,38)>;
G:=Group( (1,61,22,38)(2,62,23,39)(3,63,24,40)(4,64,17,33)(5,57,18,34)(6,58,19,35)(7,59,20,36)(8,60,21,37)(9,52,47,27)(10,53,48,28)(11,54,41,29)(12,55,42,30)(13,56,43,31)(14,49,44,32)(15,50,45,25)(16,51,46,26), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,44)(18,45)(19,46)(20,47)(21,48)(22,41)(23,42)(24,43)(25,57)(26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(32,64)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56), (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), (1,21,22,8)(2,7,23,20)(3,19,24,6)(4,5,17,18)(9,55,47,30)(10,29,48,54)(11,53,41,28)(12,27,42,52)(13,51,43,26)(14,25,44,50)(15,49,45,32)(16,31,46,56)(33,57,64,34)(35,63,58,40)(36,39,59,62)(37,61,60,38) );
G=PermutationGroup([[(1,61,22,38),(2,62,23,39),(3,63,24,40),(4,64,17,33),(5,57,18,34),(6,58,19,35),(7,59,20,36),(8,60,21,37),(9,52,47,27),(10,53,48,28),(11,54,41,29),(12,55,42,30),(13,56,43,31),(14,49,44,32),(15,50,45,25),(16,51,46,26)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,9),(8,10),(17,44),(18,45),(19,46),(20,47),(21,48),(22,41),(23,42),(24,43),(25,57),(26,58),(27,59),(28,60),(29,61),(30,62),(31,63),(32,64),(33,49),(34,50),(35,51),(36,52),(37,53),(38,54),(39,55),(40,56)], [(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)], [(1,21,22,8),(2,7,23,20),(3,19,24,6),(4,5,17,18),(9,55,47,30),(10,29,48,54),(11,53,41,28),(12,27,42,52),(13,51,43,26),(14,25,44,50),(15,49,45,32),(16,31,46,56),(33,57,64,34),(35,63,58,40),(36,39,59,62),(37,61,60,38)]])
Matrix representation of D4.2D8 ►in GL4(𝔽17) generated by
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 14 | 3 | 0 | 0 |
| 3 | 3 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 11 | 11 |
| 0 | 0 | 3 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 11 | 11 |
| 0 | 0 | 3 | 6 |
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[14,3,0,0,3,3,0,0,0,0,1,0,0,0,0,1],[13,0,0,0,0,13,0,0,0,0,11,3,0,0,11,0],[13,0,0,0,0,4,0,0,0,0,11,3,0,0,11,6] >;
D4.2D8 in GAP, Magma, Sage, TeX
D_4._2D_8
% in TeX
G:=Group("D4.2D8"); // GroupNames label
G:=SmallGroup(128,413);
// by ID
G=gap.SmallGroup(128,413);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,736,422,1123,136,2804,718,172]);
// 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^-1>;
// generators/relations
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