p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q16⋊4D4, C42.444C23, C4.1312+ (1+4), C8.9(C2×D4), C8⋊6D4⋊8C2, C2.61(D42), C8⋊D4⋊32C2, C4⋊C4.152D4, Q8⋊5D4⋊5C2, (C4×Q16)⋊31C2, Q8.24(C2×D4), D4⋊D4⋊40C2, (C2×D4).166D4, C4⋊C8.96C22, (C2×C8).91C23, C2.40(Q8○D8), C4.91(C22×D4), C8.12D4⋊23C2, C4⋊C4.216C23, (C4×C8).190C22, (C2×C4).475C24, C22⋊Q16⋊29C2, Q8.D4⋊39C2, C22⋊C4.162D4, (C2×D8).33C22, C23.317(C2×D4), (C4×D4).149C22, (C2×D4).213C23, C4⋊D4.63C22, C22⋊C8.74C22, (C2×Q8).390C23, (C2×Q16).34C22, (C4×Q8).141C22, C2.D8.222C22, C22⋊Q8.62C22, D4⋊C4.67C22, (C22×C4).325C23, (C2×SD16).50C22, C4.4D4.55C22, C22.735(C22×D4), C2.81(D8⋊C22), Q8⋊C4.157C22, (C22×Q8).332C22, (C2×M4(2)).106C22, (C2×C4).158(C2×D4), (C2×C8.C22)⋊32C2, (C2×C4○D4).190C22, SmallGroup(128,2009)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 472 in 238 conjugacy classes, 94 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×12], C22, C22 [×12], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×2], C2×C4 [×18], D4 [×13], Q8 [×4], Q8 [×10], C23 [×2], C23 [×2], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×2], M4(2) [×4], D8, SD16 [×8], Q16 [×4], Q16 [×5], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×2], C2×D4 [×4], C2×Q8 [×4], C2×Q8 [×8], C4○D4 [×6], C4×C8, C22⋊C8 [×2], D4⋊C4 [×2], Q8⋊C4 [×4], C4⋊C8, C2.D8, C4×D4, C4×D4 [×2], C4×Q8 [×2], C4⋊D4 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22⋊Q8 [×2], C4.4D4 [×2], C4.4D4 [×2], C2×M4(2) [×2], C2×D8, C2×SD16 [×4], C2×Q16 [×2], C2×Q16 [×2], C8.C22 [×8], C22×Q8 [×2], C2×C4○D4 [×2], C8⋊6D4, C4×Q16, D4⋊D4 [×2], C22⋊Q16 [×2], Q8.D4 [×2], C8⋊D4 [×2], C8.12D4, Q8⋊5D4 [×2], C2×C8.C22 [×2], Q16⋊4D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C24, C22×D4 [×2], 2+ (1+4), D42, D8⋊C22, Q8○D8, Q16⋊4D4
Generators and relations
G = < a,b,c,d | a8=c4=d2=1, b2=a4, bab-1=cac-1=a-1, dad=a3, bc=cb, bd=db, dcd=c-1 >
(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 15 5 11)(2 14 6 10)(3 13 7 9)(4 12 8 16)(17 46 21 42)(18 45 22 41)(19 44 23 48)(20 43 24 47)(25 57 29 61)(26 64 30 60)(27 63 31 59)(28 62 32 58)(33 50 37 54)(34 49 38 53)(35 56 39 52)(36 55 40 51)
(1 62 44 38)(2 61 45 37)(3 60 46 36)(4 59 47 35)(5 58 48 34)(6 57 41 33)(7 64 42 40)(8 63 43 39)(9 30 17 51)(10 29 18 50)(11 28 19 49)(12 27 20 56)(13 26 21 55)(14 25 22 54)(15 32 23 53)(16 31 24 52)
(1 50)(2 53)(3 56)(4 51)(5 54)(6 49)(7 52)(8 55)(9 35)(10 38)(11 33)(12 36)(13 39)(14 34)(15 37)(16 40)(17 59)(18 62)(19 57)(20 60)(21 63)(22 58)(23 61)(24 64)(25 48)(26 43)(27 46)(28 41)(29 44)(30 47)(31 42)(32 45)
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), (1,15,5,11)(2,14,6,10)(3,13,7,9)(4,12,8,16)(17,46,21,42)(18,45,22,41)(19,44,23,48)(20,43,24,47)(25,57,29,61)(26,64,30,60)(27,63,31,59)(28,62,32,58)(33,50,37,54)(34,49,38,53)(35,56,39,52)(36,55,40,51), (1,62,44,38)(2,61,45,37)(3,60,46,36)(4,59,47,35)(5,58,48,34)(6,57,41,33)(7,64,42,40)(8,63,43,39)(9,30,17,51)(10,29,18,50)(11,28,19,49)(12,27,20,56)(13,26,21,55)(14,25,22,54)(15,32,23,53)(16,31,24,52), (1,50)(2,53)(3,56)(4,51)(5,54)(6,49)(7,52)(8,55)(9,35)(10,38)(11,33)(12,36)(13,39)(14,34)(15,37)(16,40)(17,59)(18,62)(19,57)(20,60)(21,63)(22,58)(23,61)(24,64)(25,48)(26,43)(27,46)(28,41)(29,44)(30,47)(31,42)(32,45)>;
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), (1,15,5,11)(2,14,6,10)(3,13,7,9)(4,12,8,16)(17,46,21,42)(18,45,22,41)(19,44,23,48)(20,43,24,47)(25,57,29,61)(26,64,30,60)(27,63,31,59)(28,62,32,58)(33,50,37,54)(34,49,38,53)(35,56,39,52)(36,55,40,51), (1,62,44,38)(2,61,45,37)(3,60,46,36)(4,59,47,35)(5,58,48,34)(6,57,41,33)(7,64,42,40)(8,63,43,39)(9,30,17,51)(10,29,18,50)(11,28,19,49)(12,27,20,56)(13,26,21,55)(14,25,22,54)(15,32,23,53)(16,31,24,52), (1,50)(2,53)(3,56)(4,51)(5,54)(6,49)(7,52)(8,55)(9,35)(10,38)(11,33)(12,36)(13,39)(14,34)(15,37)(16,40)(17,59)(18,62)(19,57)(20,60)(21,63)(22,58)(23,61)(24,64)(25,48)(26,43)(27,46)(28,41)(29,44)(30,47)(31,42)(32,45) );
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)], [(1,15,5,11),(2,14,6,10),(3,13,7,9),(4,12,8,16),(17,46,21,42),(18,45,22,41),(19,44,23,48),(20,43,24,47),(25,57,29,61),(26,64,30,60),(27,63,31,59),(28,62,32,58),(33,50,37,54),(34,49,38,53),(35,56,39,52),(36,55,40,51)], [(1,62,44,38),(2,61,45,37),(3,60,46,36),(4,59,47,35),(5,58,48,34),(6,57,41,33),(7,64,42,40),(8,63,43,39),(9,30,17,51),(10,29,18,50),(11,28,19,49),(12,27,20,56),(13,26,21,55),(14,25,22,54),(15,32,23,53),(16,31,24,52)], [(1,50),(2,53),(3,56),(4,51),(5,54),(6,49),(7,52),(8,55),(9,35),(10,38),(11,33),(12,36),(13,39),(14,34),(15,37),(16,40),(17,59),(18,62),(19,57),(20,60),(21,63),(22,58),(23,61),(24,64),(25,48),(26,43),(27,46),(28,41),(29,44),(30,47),(31,42),(32,45)])
Matrix representation ►G ⊆ GL6(𝔽17)
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 3 | 3 | 7 | 0 |
0 | 0 | 14 | 3 | 0 | 10 |
0 | 0 | 0 | 7 | 14 | 3 |
0 | 0 | 7 | 0 | 14 | 14 |
16 | 0 | 0 | 0 | 0 | 0 |
0 | 16 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 16 | 0 | 0 | 0 |
0 | 0 | 0 | 16 | 0 | 0 |
0 | 16 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 13 | 4 | 0 | 1 |
0 | 0 | 4 | 4 | 16 | 0 |
0 | 0 | 0 | 16 | 13 | 4 |
0 | 0 | 1 | 0 | 4 | 4 |
0 | 1 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 14 | 14 | 0 | 0 |
0 | 0 | 14 | 3 | 0 | 0 |
0 | 0 | 0 | 0 | 14 | 14 |
0 | 0 | 0 | 0 | 14 | 3 |
G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,14,0,7,0,0,3,3,7,0,0,0,7,0,14,14,0,0,0,10,3,14],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,13,4,0,1,0,0,4,4,16,0,0,0,0,16,13,4,0,0,1,0,4,4],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,14,14,0,0,0,0,14,3,0,0,0,0,0,0,14,14,0,0,0,0,14,3] >;
Character table of Q16⋊4D4
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 8A | 8B | 8C | 8D | 8E | 8F | |
size | 1 | 1 | 1 | 1 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 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 | 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 |
ρ10 | 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 |
ρ11 | 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 |
ρ12 | 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 |
ρ13 | 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 |
ρ14 | 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 |
ρ15 | 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 |
ρ16 | 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 |
ρ17 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ18 | 2 | 2 | 2 | 2 | -2 | 2 | 0 | 0 | -2 | -2 | 2 | 2 | 0 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ19 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ20 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ21 | 2 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -2 | -2 | 2 | 2 | 0 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ23 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ25 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ (1+4) |
ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | 0 | 0 | 2√2 | 0 | 0 | symplectic lifted from Q8○D8, Schur index 2 |
ρ27 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | 0 | 0 | 2√2 | 0 | 0 | symplectic lifted from Q8○D8, Schur index 2 |
ρ28 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 4i | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
ρ29 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 4i | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
In GAP, Magma, Sage, TeX
Q_{16}\rtimes_4D_4
% in TeX
G:=Group("Q16:4D4");
// GroupNames label
G:=SmallGroup(128,2009);
// by ID
G=gap.SmallGroup(128,2009);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,723,346,248,2804,1411,375,172]);
// Polycyclic
G:=Group<a,b,c,d|a^8=c^4=d^2=1,b^2=a^4,b*a*b^-1=c*a*c^-1=a^-1,d*a*d=a^3,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations