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G = Q164D4order 128 = 27

3rd semidirect product of Q16 and D4 acting via D4/C4=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: Q164D4, C42.444C23, C4.1312+ 1+4, C2.61D42, C8.9(C2×D4), C86D48C2, C8⋊D432C2, C4⋊C4.152D4, Q85D45C2, (C4×Q16)⋊31C2, Q8.24(C2×D4), D4⋊D440C2, (C2×D4).166D4, C4⋊C8.96C22, (C2×C8).91C23, C2.40(Q8○D8), C4.91(C22×D4), C8.12D423C2, C4⋊C4.216C23, (C2×C4).475C24, (C4×C8).190C22, Q8.D439C2, C22⋊Q1629C2, C22⋊C4.162D4, (C2×D8).33C22, C23.317(C2×D4), (C2×D4).213C23, (C4×D4).149C22, C4⋊D4.63C22, C22⋊C8.74C22, (C2×Q16).34C22, (C2×Q8).390C23, (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

C1C2×C4 — Q164D4
C1C2C22C2×C4C2×Q8C22×Q8C2×C8.C22 — Q164D4
C1C2C2×C4 — Q164D4
C1C22C4×D4 — Q164D4
C1C2C2C2×C4 — Q164D4

Generators and relations for Q164D4
 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 >

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], C86D4, C4×Q16, D4⋊D4 [×2], C22⋊Q16 [×2], Q8.D4 [×2], C8⋊D4 [×2], C8.12D4, Q85D4 [×2], C2×C8.C22 [×2], Q164D4
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, Q164D4

Character table of Q164D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O8A8B8C8D8E8F
 size 11114488222244444448888444488
ρ111111111111111111111111111111    trivial
ρ21111-111111-1-1-1-1-11-111-1-11-11-1-111-1    linear of order 2
ρ31111-11-1111-1-1-1-1-11-1-1-11-111-111-1-11    linear of order 2
ρ4111111-11111111111-1-1-111-1-1-1-1-1-1-1    linear of order 2
ρ511111-11111-1-1111-1-1-1-1-1-1-111-1-11-11    linear of order 2
ρ61111-1-1111111-1-1-1-11-1-111-1-11111-1-1    linear of order 2
ρ71111-1-1-111111-1-1-1-1111-11-11-1-1-1-111    linear of order 2
ρ811111-1-1111-1-1111-1-1111-1-1-1-111-11-1    linear of order 2
ρ91111-111-111-1-11-111-111-11-1-1-111-1-11    linear of order 2
ρ101111111-11111-11-111111-1-11-1-1-1-1-1-1    linear of order 2
ρ11111111-1-11111-11-111-1-1-1-1-1-1111111    linear of order 2
ρ121111-11-1-111-1-11-111-1-1-111-111-1-111-1    linear of order 2
ρ131111-1-11-111111-11-11-1-11-11-1-1-1-1-111    linear of order 2
ρ1411111-11-111-1-1-11-1-1-1-1-1-1111-111-11-1    linear of order 2
ρ1511111-1-1-111-1-1-11-1-1-111111-11-1-11-11    linear of order 2
ρ161111-1-1-1-111111-11-1111-1-1111111-1-1    linear of order 2
ρ172-22-200002-20020-200-22000002-2000    orthogonal lifted from D4
ρ182222-2200-2-222020-2-2000000000000    orthogonal lifted from D4
ρ1922222200-2-2-2-20-20-22000000000000    orthogonal lifted from D4
ρ202-22-200002-200-20200-2200000-22000    orthogonal lifted from D4
ρ2122222-200-2-2220-202-2000000000000    orthogonal lifted from D4
ρ222-22-200002-200-202002-2000002-2000    orthogonal lifted from D4
ρ232222-2-200-2-2-2-202022000000000000    orthogonal lifted from D4
ρ242-22-200002-20020-2002-200000-22000    orthogonal lifted from D4
ρ254-44-40000-440000000000000000000    orthogonal lifted from 2+ 1+4
ρ2644-4-400000000000000000002200-2200    symplectic lifted from Q8○D8, Schur index 2
ρ2744-4-40000000000000000000-22002200    symplectic lifted from Q8○D8, Schur index 2
ρ284-4-440000004i-4i00000000000000000    complex lifted from D8⋊C22
ρ294-4-44000000-4i4i00000000000000000    complex lifted from D8⋊C22

Smallest permutation representation of Q164D4
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 9 5 13)(2 16 6 12)(3 15 7 11)(4 14 8 10)(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 52 37 56)(34 51 38 55)(35 50 39 54)(36 49 40 53)
(1 62 44 40)(2 61 45 39)(3 60 46 38)(4 59 47 37)(5 58 48 36)(6 57 41 35)(7 64 42 34)(8 63 43 33)(9 32 23 53)(10 31 24 52)(11 30 17 51)(12 29 18 50)(13 28 19 49)(14 27 20 56)(15 26 21 55)(16 25 22 54)
(1 50)(2 53)(3 56)(4 51)(5 54)(6 49)(7 52)(8 55)(9 39)(10 34)(11 37)(12 40)(13 35)(14 38)(15 33)(16 36)(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), (1,9,5,13)(2,16,6,12)(3,15,7,11)(4,14,8,10)(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)(33,52,37,56)(34,51,38,55)(35,50,39,54)(36,49,40,53), (1,62,44,40)(2,61,45,39)(3,60,46,38)(4,59,47,37)(5,58,48,36)(6,57,41,35)(7,64,42,34)(8,63,43,33)(9,32,23,53)(10,31,24,52)(11,30,17,51)(12,29,18,50)(13,28,19,49)(14,27,20,56)(15,26,21,55)(16,25,22,54), (1,50)(2,53)(3,56)(4,51)(5,54)(6,49)(7,52)(8,55)(9,39)(10,34)(11,37)(12,40)(13,35)(14,38)(15,33)(16,36)(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), (1,9,5,13)(2,16,6,12)(3,15,7,11)(4,14,8,10)(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)(33,52,37,56)(34,51,38,55)(35,50,39,54)(36,49,40,53), (1,62,44,40)(2,61,45,39)(3,60,46,38)(4,59,47,37)(5,58,48,36)(6,57,41,35)(7,64,42,34)(8,63,43,33)(9,32,23,53)(10,31,24,52)(11,30,17,51)(12,29,18,50)(13,28,19,49)(14,27,20,56)(15,26,21,55)(16,25,22,54), (1,50)(2,53)(3,56)(4,51)(5,54)(6,49)(7,52)(8,55)(9,39)(10,34)(11,37)(12,40)(13,35)(14,38)(15,33)(16,36)(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)], [(1,9,5,13),(2,16,6,12),(3,15,7,11),(4,14,8,10),(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),(33,52,37,56),(34,51,38,55),(35,50,39,54),(36,49,40,53)], [(1,62,44,40),(2,61,45,39),(3,60,46,38),(4,59,47,37),(5,58,48,36),(6,57,41,35),(7,64,42,34),(8,63,43,33),(9,32,23,53),(10,31,24,52),(11,30,17,51),(12,29,18,50),(13,28,19,49),(14,27,20,56),(15,26,21,55),(16,25,22,54)], [(1,50),(2,53),(3,56),(4,51),(5,54),(6,49),(7,52),(8,55),(9,39),(10,34),(11,37),(12,40),(13,35),(14,38),(15,33),(16,36),(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 of Q164D4 in GL6(𝔽17)

100000
010000
003370
00143010
0007143
00701414
,
1600000
0160000
000010
000001
0016000
0001600
,
0160000
100000
0013401
0044160
00016134
001044
,
010000
100000
00141400
0014300
00001414
0000143

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] >;

Q164D4 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

Export

Character table of Q164D4 in TeX

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