Copied to
clipboard

G = Q16⋊C4order 64 = 26

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

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

Aliases: Q163C4, C42.10C22, C8.5(C2×C4), C2.16(C4×D4), Q8.2(C2×C4), (C4×Q8).3C2, C8⋊C4.1C2, C4.Q8.2C2, C4.5(C4○D4), (C2×C4).102D4, (C2×Q16).6C2, C4⋊C4.54C22, (C2×C8).50C22, (C2×C4).77C23, C4.13(C22×C4), Q8⋊C4.6C2, C22.55(C2×D4), C2.5(C8.C22), (C2×Q8).47C22, SmallGroup(64,122)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — Q16⋊C4
C1C2C22C2×C4C42C4×Q8 — Q16⋊C4
C1C2C4 — Q16⋊C4
C1C22C42 — Q16⋊C4
C1C2C2C2×C4 — Q16⋊C4

Generators and relations for Q16⋊C4
 G = < a,b,c | a8=c4=1, b2=a4, bab-1=a-1, cac-1=a5, bc=cb >

2C4
2C4
2C4
2C4
2C4
2C4
4C4
4C4
2C2×C4
2C2×C4
2Q8
2Q8
2C2×C4
2C2×C4
2C8
2C42
2C4⋊C4
2C4⋊C4
2C42

Character table of Q16⋊C4

 class 12A2B2C4A4B4C4D4E4F4G4H4I4J4K4L4M4N8A8B8C8D
 size 1111222222444444444444
ρ11111111111111111111111    trivial
ρ21111-1-111-1-1-111-1-111-1-11-11    linear of order 2
ρ31111-1-111-1-11-1-111-1-11-11-11    linear of order 2
ρ41111111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ5111111111111-1-111-1-1-1-1-1-1    linear of order 2
ρ61111-1-111-1-1-11-11-11-111-11-1    linear of order 2
ρ71111-1-111-1-11-11-11-11-11-11-1    linear of order 2
ρ81111111111-1-111-1-111-1-1-1-1    linear of order 2
ρ91-11-1-ii1-1-iii-1-1-i-i11i-i1i-1    linear of order 4
ρ101-11-1-ii1-1-iii-11i-i1-1-ii-1-i1    linear of order 4
ρ111-11-1i-i1-1i-ii11-i-i-1-1ii1-i-1    linear of order 4
ρ121-11-1i-i1-1i-ii1-1i-i-11-i-i-1i1    linear of order 4
ρ131-11-1i-i1-1i-i-i-1-1ii11-ii1-i-1    linear of order 4
ρ141-11-1i-i1-1i-i-i-11-ii1-1i-i-1i1    linear of order 4
ρ151-11-1-ii1-1-ii-i11ii-1-1-i-i1i-1    linear of order 4
ρ161-11-1-ii1-1-ii-i1-1-ii-11ii-1-i1    linear of order 4
ρ172222-22-2-22-2000000000000    orthogonal lifted from D4
ρ1822222-2-2-2-22000000000000    orthogonal lifted from D4
ρ192-22-2-2i-2i-222i2i000000000000    complex lifted from C4○D4
ρ202-22-22i2i-22-2i-2i000000000000    complex lifted from C4○D4
ρ2144-4-4000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ224-4-44000000000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of Q16⋊C4
Regular action 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 13 5 9)(2 12 6 16)(3 11 7 15)(4 10 8 14)(17 55 21 51)(18 54 22 50)(19 53 23 49)(20 52 24 56)(25 45 29 41)(26 44 30 48)(27 43 31 47)(28 42 32 46)(33 57 37 61)(34 64 38 60)(35 63 39 59)(36 62 40 58)
(1 31 64 51)(2 28 57 56)(3 25 58 53)(4 30 59 50)(5 27 60 55)(6 32 61 52)(7 29 62 49)(8 26 63 54)(9 43 34 21)(10 48 35 18)(11 45 36 23)(12 42 37 20)(13 47 38 17)(14 44 39 22)(15 41 40 19)(16 46 33 24)

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

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

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

Q16⋊C4 is a maximal subgroup of
C42.354C23  C42.355C23  C42.359C23  C42.361C23  C42.385C23  C42.389C23  C42.390C23  Q169D4  Q1610D4  C42.47C23  C42.48C23  C42.51C23  C42.52C23  C42.58C23  C42.63C23  C42.510C23  C42.512C23  C42.515C23  C42.518C23  Q164Q8  Q165Q8  C42.73C23  C42.531C23  CSU2(𝔽3)⋊C4
 C42.D2p: Q32⋊C4  C42.383D4  C4×C8.C22  C42.229D4  C42.231D4  C42.234D4  C42.235D4  C42.258D4 ...
 (Cp×Q16)⋊C4: C42.279C23  Q16⋊Dic3  Q16⋊Dic5  Dic20⋊C4  Q16⋊Dic7 ...
 C2p.(C4×D4): C42.276C23  C42.281C23  C3⋊Q16⋊C4  Dic129C4  C5⋊Q165C4  Dic2015C4  C7⋊Q16⋊C4  Dic289C4 ...
Q16⋊C4 is a maximal quotient of
Q165C8  Q8.M4(2)  C8.M4(2)  C8⋊C42  C2.D84C4  C4.(C4×Q8)
 C42.D2p: Q8⋊C42  Dic12⋊C4  C42.59D6  Dic209C4  C42.59D10  Dic28⋊C4  C42.59D14 ...
 (Cp×Q16)⋊C4: (C2×C4)⋊9Q16  (C2×Q16)⋊10C4  Q16⋊Dic3  Q16⋊Dic5  Dic20⋊C4  Q16⋊Dic7 ...
 C2p.(C4×D4): Q8⋊C4⋊C4  C4.68(C4×D4)  C2.(C8⋊D4)  C3⋊Q16⋊C4  Dic129C4  C5⋊Q165C4  Dic2015C4  C7⋊Q16⋊C4 ...

Matrix representation of Q16⋊C4 in GL6(𝔽17)

400000
7130000
004080
004044
00013130
0000130
,
6150000
9110000
001020
0000116
00160160
00161160
,
400000
040000
0011500
0011600
000101
00161160

G:=sub<GL(6,GF(17))| [4,7,0,0,0,0,0,13,0,0,0,0,0,0,4,4,0,0,0,0,0,0,13,0,0,0,8,4,13,13,0,0,0,4,0,0],[6,9,0,0,0,0,15,11,0,0,0,0,0,0,1,0,16,16,0,0,0,0,0,1,0,0,2,1,16,16,0,0,0,16,0,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,1,0,16,0,0,15,16,1,1,0,0,0,0,0,16,0,0,0,0,1,0] >;

Q16⋊C4 in GAP, Magma, Sage, TeX

Q_{16}\rtimes C_4
% in TeX

G:=Group("Q16:C4");
// GroupNames label

G:=SmallGroup(64,122);
// by ID

G=gap.SmallGroup(64,122);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,199,650,86,963,489,117]);
// Polycyclic

G:=Group<a,b,c|a^8=c^4=1,b^2=a^4,b*a*b^-1=a^-1,c*a*c^-1=a^5,b*c=c*b>;
// generators/relations

Export

Subgroup lattice of Q16⋊C4 in TeX
Character table of Q16⋊C4 in TeX

׿
×
𝔽