p-group, metabelian, nilpotent (class 4), monomial
Aliases: D4.4D8, Q8.4D8, C16.21D4, M4(2).35D4, M5(2).27C22, (C2xQ32):9C2, C4.42(C2xD8), D4oC16.1C2, C4oD4.27D4, C8.5(C4oD4), C8.106(C2xD4), D4.5D4.C2, C8.4Q8:6C2, C8.17D4:7C2, C8oD4.8C22, C2.25(C8:7D4), C4.97(C4:D4), (C2xC16).27C22, (C2xC8).239C23, C22.7(C4oD8), C8.C4.6C22, (C2xQ16).48C22, (C2xC4).44(C2xD4), SmallGroup(128,954)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4.4D8
G = < a,b,c,d | a4=b2=1, c8=d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2c7 >
Subgroups: 148 in 70 conjugacy classes, 30 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, C16, C16, C2xC8, C2xC8, M4(2), M4(2), SD16, Q16, C2xQ8, C4oD4, C4.10D4, C8.C4, C2xC16, C2xC16, M5(2), M5(2), Q32, C8oD4, C2xQ16, C8.C22, C8.17D4, C8.4Q8, D4.5D4, D4oC16, C2xQ32, D4.4D8
Quotients: C1, C2, C22, D4, C23, D8, C2xD4, C4oD4, C4:D4, C2xD8, C4oD8, C8:7D4, D4.4D8
Character table of D4.4D8
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 16I | 16J | |
| size | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 16 | 16 | 2 | 2 | 4 | 4 | 4 | 16 | 16 | 2 | 2 | 2 | 2 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | √2 | -√2 | √2 | -√2 | √2 | -√2 | orthogonal lifted from D8 |
| ρ14 | 2 | 2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | -√2 | √2 | -√2 | √2 | -√2 | √2 | orthogonal lifted from D8 |
| ρ15 | 2 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | √2 | √2 | -√2 | √2 | -√2 | -√2 | orthogonal lifted from D8 |
| ρ16 | 2 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | -√2 | -√2 | √2 | -√2 | √2 | √2 | orthogonal lifted from D8 |
| ρ17 | 2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | -2i | 2i | 2i | 0 | complex lifted from C4oD4 |
| ρ18 | 2 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | √2 | -√2 | √2 | -√2 | -√2 | √-2 | -√-2 | -√-2 | √-2 | √2 | complex lifted from C4oD8 |
| ρ19 | 2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 2i | -2i | -2i | 0 | complex lifted from C4oD4 |
| ρ20 | 2 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | -√2 | √2 | -√2 | √2 | √2 | -√-2 | √-2 | √-2 | -√-2 | -√2 | complex lifted from C4oD8 |
| ρ21 | 2 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | √2 | -√2 | √2 | -√2 | -√2 | -√-2 | √-2 | √-2 | -√-2 | √2 | complex lifted from C4oD8 |
| ρ22 | 2 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | -√2 | √2 | -√2 | √2 | √2 | √-2 | -√-2 | -√-2 | √-2 | -√2 | complex lifted from C4oD8 |
| ρ23 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | -2ζ167+2ζ16 | -2ζ165+2ζ163 | 2ζ167-2ζ16 | 2ζ165-2ζ163 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ24 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 2ζ165-2ζ163 | -2ζ167+2ζ16 | -2ζ165+2ζ163 | 2ζ167-2ζ16 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 2ζ167-2ζ16 | 2ζ165-2ζ163 | -2ζ167+2ζ16 | -2ζ165+2ζ163 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | -2ζ165+2ζ163 | 2ζ167-2ζ16 | 2ζ165-2ζ163 | -2ζ167+2ζ16 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 64 points
Generators in S64
(1 44 9 36)(2 45 10 37)(3 46 11 38)(4 47 12 39)(5 48 13 40)(6 33 14 41)(7 34 15 42)(8 35 16 43)(17 52 25 60)(18 53 26 61)(19 54 27 62)(20 55 28 63)(21 56 29 64)(22 57 30 49)(23 58 31 50)(24 59 32 51)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 33)(15 34)(16 35)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)
(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 24 9 32)(2 23 10 31)(3 22 11 30)(4 21 12 29)(5 20 13 28)(6 19 14 27)(7 18 15 26)(8 17 16 25)(33 62 41 54)(34 61 42 53)(35 60 43 52)(36 59 44 51)(37 58 45 50)(38 57 46 49)(39 56 47 64)(40 55 48 63)
G:=sub<Sym(64)| (1,44,9,36)(2,45,10,37)(3,46,11,38)(4,47,12,39)(5,48,13,40)(6,33,14,41)(7,34,15,42)(8,35,16,43)(17,52,25,60)(18,53,26,61)(19,54,27,62)(20,55,28,63)(21,56,29,64)(22,57,30,49)(23,58,31,50)(24,59,32,51), (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,33)(15,34)(16,35)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (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), (1,24,9,32)(2,23,10,31)(3,22,11,30)(4,21,12,29)(5,20,13,28)(6,19,14,27)(7,18,15,26)(8,17,16,25)(33,62,41,54)(34,61,42,53)(35,60,43,52)(36,59,44,51)(37,58,45,50)(38,57,46,49)(39,56,47,64)(40,55,48,63)>;
G:=Group( (1,44,9,36)(2,45,10,37)(3,46,11,38)(4,47,12,39)(5,48,13,40)(6,33,14,41)(7,34,15,42)(8,35,16,43)(17,52,25,60)(18,53,26,61)(19,54,27,62)(20,55,28,63)(21,56,29,64)(22,57,30,49)(23,58,31,50)(24,59,32,51), (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,33)(15,34)(16,35)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32), (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), (1,24,9,32)(2,23,10,31)(3,22,11,30)(4,21,12,29)(5,20,13,28)(6,19,14,27)(7,18,15,26)(8,17,16,25)(33,62,41,54)(34,61,42,53)(35,60,43,52)(36,59,44,51)(37,58,45,50)(38,57,46,49)(39,56,47,64)(40,55,48,63) );
G=PermutationGroup([[(1,44,9,36),(2,45,10,37),(3,46,11,38),(4,47,12,39),(5,48,13,40),(6,33,14,41),(7,34,15,42),(8,35,16,43),(17,52,25,60),(18,53,26,61),(19,54,27,62),(20,55,28,63),(21,56,29,64),(22,57,30,49),(23,58,31,50),(24,59,32,51)], [(1,36),(2,37),(3,38),(4,39),(5,40),(6,41),(7,42),(8,43),(9,44),(10,45),(11,46),(12,47),(13,48),(14,33),(15,34),(16,35),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32)], [(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)], [(1,24,9,32),(2,23,10,31),(3,22,11,30),(4,21,12,29),(5,20,13,28),(6,19,14,27),(7,18,15,26),(8,17,16,25),(33,62,41,54),(34,61,42,53),(35,60,43,52),(36,59,44,51),(37,58,45,50),(38,57,46,49),(39,56,47,64),(40,55,48,63)]])
Matrix representation of D4.4D8 ►in GL4(F17) generated by
| 1 | 0 | 15 | 0 |
| 0 | 0 | 16 | 1 |
| 1 | 0 | 16 | 0 |
| 1 | 16 | 16 | 0 |
| 1 | 0 | 15 | 0 |
| 0 | 0 | 16 | 1 |
| 0 | 0 | 16 | 0 |
| 0 | 1 | 16 | 0 |
| 2 | 8 | 0 | 0 |
| 13 | 10 | 0 | 0 |
| 0 | 4 | 6 | 13 |
| 13 | 4 | 4 | 6 |
| 0 | 9 | 6 | 8 |
| 0 | 10 | 16 | 7 |
| 7 | 9 | 0 | 0 |
| 1 | 16 | 16 | 7 |
G:=sub<GL(4,GF(17))| [1,0,1,1,0,0,0,16,15,16,16,16,0,1,0,0],[1,0,0,0,0,0,0,1,15,16,16,16,0,1,0,0],[2,13,0,13,8,10,4,4,0,0,6,4,0,0,13,6],[0,0,7,1,9,10,9,16,6,16,0,16,8,7,0,7] >;
D4.4D8 in GAP, Magma, Sage, TeX
D_4._4D_8
% in TeX
G:=Group("D4.4D8"); // GroupNames label
G:=SmallGroup(128,954);
// by ID
G=gap.SmallGroup(128,954);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,288,422,1123,360,2804,718,172,4037,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=1,c^8=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^7>;
// generators/relations
Export