p-group, metabelian, nilpotent (class 4), monomial
Aliases: D4.4D8, Q8.4D8, C16.21D4, M4(2).35D4, M5(2).27C22, (C2×Q32)⋊9C2, C4.42(C2×D8), D4○C16.1C2, C4○D4.27D4, C8.5(C4○D4), C8.106(C2×D4), D4.5D4.C2, C8.4Q8⋊6C2, C8.17D4⋊7C2, C8○D4.8C22, C2.25(C8⋊7D4), C4.97(C4⋊D4), (C2×C16).27C22, (C2×C8).239C23, C22.7(C4○D8), C8.C4.6C22, (C2×Q16).48C22, (C2×C4).44(C2×D4), 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, C2×C4, C2×C4, D4, D4, Q8, Q8, C16, C16, C2×C8, C2×C8, M4(2), M4(2), SD16, Q16, C2×Q8, C4○D4, C4.10D4, C8.C4, C2×C16, C2×C16, M5(2), M5(2), Q32, C8○D4, C2×Q16, C8.C22, C8.17D4, C8.4Q8, D4.5D4, D4○C16, C2×Q32, D4.4D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C4○D4, C4⋊D4, C2×D8, C4○D8, 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 C4○D4 |
| 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 C4○D8 |
| 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 C4○D4 |
| 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 C4○D8 |
| 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 C4○D8 |
| 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 C4○D8 |
| 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
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