p-group, metabelian, nilpotent (class 4), monomial
Aliases: D8⋊2D4, C4⋊2D16, C42.136D4, C4⋊C16⋊5C2, (C4×D8)⋊2C2, (C2×D16)⋊4C2, C8⋊4D4⋊6C2, (C2×C8).65D4, C2.5(C2×D16), C8.70(C2×D4), C2.D16⋊6C2, (C2×C4).147D8, C8.88(C4○D4), (C4×C8).61C22, (C2×C16).4C22, (C2×D8).2C22, C2.18(C4⋊D8), C4.49(C4⋊D4), C4.14(C8⋊C22), C2.9(C16⋊C22), (C2×C8).514C23, C22.100(C2×D8), C2.D8.156C22, (C2×C4).782(C2×D4), SmallGroup(128,938)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D8⋊2D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=dad=a-1, ac=ca, bc=cb, dbd=a5b, dcd=c-1 >
Subgroups: 316 in 93 conjugacy classes, 34 normal (22 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, C23, C16, C42, C22⋊C4, C4⋊C4, C2×C8, D8, D8, C22×C4, C2×D4, C4×C8, D4⋊C4, C2.D8, C2×C16, D16, C4×D4, C4⋊1D4, C2×D8, C2×D8, C2×D8, C2.D16, C4⋊C16, C4×D8, C8⋊4D4, C2×D16, D8⋊2D4
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C4○D4, D16, C4⋊D4, C2×D8, C8⋊C22, C4⋊D8, C2×D16, C16⋊C22, D8⋊2D4
Character table of D8⋊2D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 8A | 8B | 8C | 8D | 8E | 8F | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 16 | 16 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 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 | 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 | 0 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | √2 | -√2 | √2 | √2 | -√2 | -√2 | -√2 | orthogonal lifted from D8 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | -√2 | √2 | √2 | -√2 | orthogonal lifted from D8 |
| ρ15 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | √2 | -√2 | -√2 | √2 | orthogonal lifted from D8 |
| ρ16 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | -√2 | √2 | -√2 | -√2 | √2 | √2 | √2 | orthogonal lifted from D8 |
| ρ17 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | √2 | √2 | -√2 | -√2 | √2 | -√2 | -ζ167+ζ16 | -ζ167+ζ16 | -ζ165+ζ163 | ζ167-ζ16 | ζ167-ζ16 | -ζ165+ζ163 | ζ165-ζ163 | ζ165-ζ163 | orthogonal lifted from D16 |
| ρ18 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -√2 | -√2 | √2 | √2 | -√2 | √2 | -ζ165+ζ163 | -ζ165+ζ163 | ζ167-ζ16 | ζ165-ζ163 | ζ165-ζ163 | ζ167-ζ16 | -ζ167+ζ16 | -ζ167+ζ16 | orthogonal lifted from D16 |
| ρ19 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | √2 | √2 | -√2 | -√2 | -√2 | √2 | ζ167-ζ16 | -ζ167+ζ16 | -ζ165+ζ163 | ζ167-ζ16 | -ζ167+ζ16 | ζ165-ζ163 | -ζ165+ζ163 | ζ165-ζ163 | orthogonal lifted from D16 |
| ρ20 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | √2 | √2 | -√2 | -√2 | √2 | -√2 | ζ167-ζ16 | ζ167-ζ16 | ζ165-ζ163 | -ζ167+ζ16 | -ζ167+ζ16 | ζ165-ζ163 | -ζ165+ζ163 | -ζ165+ζ163 | orthogonal lifted from D16 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | -√2 | -√2 | √2 | √2 | √2 | -√2 | -ζ165+ζ163 | ζ165-ζ163 | -ζ167+ζ16 | -ζ165+ζ163 | ζ165-ζ163 | ζ167-ζ16 | -ζ167+ζ16 | ζ167-ζ16 | orthogonal lifted from D16 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | -√2 | -√2 | √2 | √2 | √2 | -√2 | ζ165-ζ163 | -ζ165+ζ163 | ζ167-ζ16 | ζ165-ζ163 | -ζ165+ζ163 | -ζ167+ζ16 | ζ167-ζ16 | -ζ167+ζ16 | orthogonal lifted from D16 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -√2 | -√2 | √2 | √2 | -√2 | √2 | ζ165-ζ163 | ζ165-ζ163 | -ζ167+ζ16 | -ζ165+ζ163 | -ζ165+ζ163 | -ζ167+ζ16 | ζ167-ζ16 | ζ167-ζ16 | orthogonal lifted from D16 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | √2 | √2 | -√2 | -√2 | -√2 | √2 | -ζ167+ζ16 | ζ167-ζ16 | ζ165-ζ163 | -ζ167+ζ16 | ζ167-ζ16 | -ζ165+ζ163 | ζ165-ζ163 | -ζ165+ζ163 | orthogonal lifted from D16 |
| ρ25 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | -2i | 2i | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ26 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 2i | -2i | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ27 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 4 | 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 C16⋊C22 |
| 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 C16⋊C22 |
►On 64 points
Generators in S64
(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 28)(2 27)(3 26)(4 25)(5 32)(6 31)(7 30)(8 29)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 24)(16 23)(33 62)(34 61)(35 60)(36 59)(37 58)(38 57)(39 64)(40 63)(41 54)(42 53)(43 52)(44 51)(45 50)(46 49)(47 56)(48 55)
(1 43 11 35)(2 44 12 36)(3 45 13 37)(4 46 14 38)(5 47 15 39)(6 48 16 40)(7 41 9 33)(8 42 10 34)(17 57 25 49)(18 58 26 50)(19 59 27 51)(20 60 28 52)(21 61 29 53)(22 62 30 54)(23 63 31 55)(24 64 32 56)
(1 11)(2 10)(3 9)(4 16)(5 15)(6 14)(7 13)(8 12)(17 26)(18 25)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(33 37)(34 36)(38 40)(41 45)(42 44)(46 48)(49 50)(51 56)(52 55)(53 54)(57 58)(59 64)(60 63)(61 62)
G:=sub<Sym(64)| (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), (1,28)(2,27)(3,26)(4,25)(5,32)(6,31)(7,30)(8,29)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,24)(16,23)(33,62)(34,61)(35,60)(36,59)(37,58)(38,57)(39,64)(40,63)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,56)(48,55), (1,43,11,35)(2,44,12,36)(3,45,13,37)(4,46,14,38)(5,47,15,39)(6,48,16,40)(7,41,9,33)(8,42,10,34)(17,57,25,49)(18,58,26,50)(19,59,27,51)(20,60,28,52)(21,61,29,53)(22,62,30,54)(23,63,31,55)(24,64,32,56), (1,11)(2,10)(3,9)(4,16)(5,15)(6,14)(7,13)(8,12)(17,26)(18,25)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(33,37)(34,36)(38,40)(41,45)(42,44)(46,48)(49,50)(51,56)(52,55)(53,54)(57,58)(59,64)(60,63)(61,62)>;
G:=Group( (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), (1,28)(2,27)(3,26)(4,25)(5,32)(6,31)(7,30)(8,29)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,24)(16,23)(33,62)(34,61)(35,60)(36,59)(37,58)(38,57)(39,64)(40,63)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,56)(48,55), (1,43,11,35)(2,44,12,36)(3,45,13,37)(4,46,14,38)(5,47,15,39)(6,48,16,40)(7,41,9,33)(8,42,10,34)(17,57,25,49)(18,58,26,50)(19,59,27,51)(20,60,28,52)(21,61,29,53)(22,62,30,54)(23,63,31,55)(24,64,32,56), (1,11)(2,10)(3,9)(4,16)(5,15)(6,14)(7,13)(8,12)(17,26)(18,25)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(33,37)(34,36)(38,40)(41,45)(42,44)(46,48)(49,50)(51,56)(52,55)(53,54)(57,58)(59,64)(60,63)(61,62) );
G=PermutationGroup([[(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)], [(1,28),(2,27),(3,26),(4,25),(5,32),(6,31),(7,30),(8,29),(9,22),(10,21),(11,20),(12,19),(13,18),(14,17),(15,24),(16,23),(33,62),(34,61),(35,60),(36,59),(37,58),(38,57),(39,64),(40,63),(41,54),(42,53),(43,52),(44,51),(45,50),(46,49),(47,56),(48,55)], [(1,43,11,35),(2,44,12,36),(3,45,13,37),(4,46,14,38),(5,47,15,39),(6,48,16,40),(7,41,9,33),(8,42,10,34),(17,57,25,49),(18,58,26,50),(19,59,27,51),(20,60,28,52),(21,61,29,53),(22,62,30,54),(23,63,31,55),(24,64,32,56)], [(1,11),(2,10),(3,9),(4,16),(5,15),(6,14),(7,13),(8,12),(17,26),(18,25),(19,32),(20,31),(21,30),(22,29),(23,28),(24,27),(33,37),(34,36),(38,40),(41,45),(42,44),(46,48),(49,50),(51,56),(52,55),(53,54),(57,58),(59,64),(60,63),(61,62)]])
Matrix representation of D8⋊2D4 ►in GL4(𝔽17) generated by
| 3 | 14 | 0 | 0 |
| 3 | 3 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 4 | 6 | 0 | 0 |
| 6 | 13 | 0 | 0 |
| 0 | 0 | 6 | 4 |
| 0 | 0 | 4 | 11 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 7 | 16 |
| 0 | 0 | 16 | 10 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 3 | 1 |
G:=sub<GL(4,GF(17))| [3,3,0,0,14,3,0,0,0,0,16,0,0,0,0,16],[4,6,0,0,6,13,0,0,0,0,6,4,0,0,4,11],[16,0,0,0,0,16,0,0,0,0,7,16,0,0,16,10],[1,0,0,0,0,16,0,0,0,0,16,3,0,0,0,1] >;
D8⋊2D4 in GAP, Magma, Sage, TeX
D_8\rtimes_2D_4
% in TeX
G:=Group("D8:2D4"); // GroupNames label
G:=SmallGroup(128,938);
// by ID
G=gap.SmallGroup(128,938);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,64,422,1684,438,242,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a^5*b,d*c*d=c^-1>;
// generators/relations
Export