Copied to
clipboard

G = C4×Q16order 64 = 26

Direct product of C4 and Q16

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

Aliases: C4×Q16, C42.73C22, (C4×C8).6C2, C8.10(C2×C4), C43(C2.D8), C2.14(C4×D4), (C2×C4).53D4, Q8.1(C2×C4), (C4×Q8).2C2, C2.3(C2×Q16), C2.D8.9C2, C4.3(C4○D4), C2.5(C4○D8), (C2×Q16).7C2, C43(Q8⋊C4), C4⋊C4.52C22, (C2×C4).75C23, C4.11(C22×C4), (C2×C8).64C22, Q8⋊C4.8C2, C22.53(C2×D4), (C2×Q8).45C22, SmallGroup(64,120)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C4×Q16
C1C2C22C2×C4C42C4×Q8 — C4×Q16
C1C2C4 — C4×Q16
C1C2×C4C42 — C4×Q16
C1C2C2C2×C4 — C4×Q16

Generators and relations for C4×Q16
 G = < a,b,c | a4=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >

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

Character table of C4×Q16

 class 12A2B2C4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P8A8B8C8D8E8F8G8H
 size 1111111122224444444422222222
ρ11111111111111111111111111111    trivial
ρ21111-1-1-1-111-1-1-111-1-1-111-11-11-11-11    linear of order 2
ρ31111-1-1-1-111-1-11-1-1111-1-1-11-11-11-11    linear of order 2
ρ4111111111111-1-1-1-1-1-1-1-111111111    linear of order 2
ρ51111-1-1-1-111-1-1-11-111-11-11-11-11-11-1    linear of order 2
ρ611111111111111-1-1-111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ71111-1-1-1-111-1-11-11-1-11-111-11-11-11-1    linear of order 2
ρ8111111111111-1-1111-1-11-1-1-1-1-1-1-1-1    linear of order 2
ρ91-1-11-i-iii-11i-i1ii-11-1-i-ii-1i-1-i1-i1    linear of order 4
ρ101-1-11-i-iii-11i-i1i-i1-1-1-ii-i1-i1i-1i-1    linear of order 4
ρ111-1-11ii-i-i-11-ii-1i-i-111-iii1i1-i-1-i-1    linear of order 4
ρ121-1-11ii-i-i-11-ii-1ii1-11-i-i-i-1-i-1i1i1    linear of order 4
ρ131-1-11-i-iii-11i-i-1-i-i1-11iii-1i-1-i1-i1    linear of order 4
ρ141-1-11-i-iii-11i-i-1-ii-111i-i-i1-i1i-1i-1    linear of order 4
ρ151-1-11ii-i-i-11-ii1-ii1-1-1i-ii1i1-i-1-i-1    linear of order 4
ρ161-1-11ii-i-i-11-ii1-i-i-11-1ii-i-1-i-1i1i1    linear of order 4
ρ172222-2-2-2-2-2-2220000000000000000    orthogonal lifted from D4
ρ1822222222-2-2-2-20000000000000000    orthogonal lifted from D4
ρ192-22-2-222-2000000000000-222-22-2-22    symplectic lifted from Q16, Schur index 2
ρ202-22-2-222-20000000000002-2-22-222-2    symplectic lifted from Q16, Schur index 2
ρ212-22-22-2-2200000000000022-2-2-2-222    symplectic lifted from Q16, Schur index 2
ρ222-22-22-2-22000000000000-2-22222-2-2    symplectic lifted from Q16, Schur index 2
ρ2322-2-2-2i2i-2i2i000000000000-2-2--22-2-2--22    complex lifted from C4○D8
ρ242-2-22-2i-2i2i2i2-2-2i2i0000000000000000    complex lifted from C4○D4
ρ2522-2-22i-2i2i-2i000000000000--2-2-22--2-2-22    complex lifted from C4○D8
ρ2622-2-22i-2i2i-2i000000000000-22--2-2-22--2-2    complex lifted from C4○D8
ρ2722-2-2-2i2i-2i2i000000000000--22-2-2--22-2-2    complex lifted from C4○D8
ρ282-2-222i2i-2i-2i2-22i-2i0000000000000000    complex lifted from C4○D4

Smallest permutation representation of C4×Q16
Regular action on 64 points
Generators in S64
(1 32 60 51)(2 25 61 52)(3 26 62 53)(4 27 63 54)(5 28 64 55)(6 29 57 56)(7 30 58 49)(8 31 59 50)(9 48 37 18)(10 41 38 19)(11 42 39 20)(12 43 40 21)(13 44 33 22)(14 45 34 23)(15 46 35 24)(16 47 36 17)
(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 12 5 16)(2 11 6 15)(3 10 7 14)(4 9 8 13)(17 51 21 55)(18 50 22 54)(19 49 23 53)(20 56 24 52)(25 42 29 46)(26 41 30 45)(27 48 31 44)(28 47 32 43)(33 63 37 59)(34 62 38 58)(35 61 39 57)(36 60 40 64)

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

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

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

C4×Q16 is a maximal subgroup of
Q161C8  Q16⋊C8  Q165C8  C89Q16  Q8.M4(2)  C86Q16  C8.M4(2)  Q162D4  Q16.4D4  Q16.5D4  Q16⋊Q8  C4.Q32  Q16.Q8  C42.384D4  C42.224D4  C42.451D4  C42.226D4  C42.354C23  C42.355C23  C42.358C23  C42.361C23  C42.308D4  C42.367D4  C42.256D4  C42.387C23  C42.389C23  C42.390C23  Q164D4  Q165D4  Q1612D4  Q1613D4  D45Q16  C42.465C23  C42.469C23  C42.476C23  C42.477C23  C42.482C23  C42.485C23  D46Q16  C42.491C23  C42.493C23  C42.497C23  Q85Q16  C42.505C23  C42.506C23  C42.515C23  C42.516C23  C42.518C23  Q166Q8  Q164Q8  Q165Q8  C42.527C23  Q86Q16  C42.530C23  C42.75C23  C42.532C23
 C2p.(C4×D4): SD323C4  Q324C4  C42.276C23  C42.279C23  C42.280C23  Dic34Q16  Dic35Q16  Dic54Q16 ...
C4×Q16 is a maximal quotient of
C89Q16  C86Q16  C2.D85C4  C85(C4⋊C4)
 C2p.(C4×D4): Q8⋊(C4⋊C4)  (C2×C4)⋊9Q16  C2.(C4×Q16)  C2.(C88D4)  (C2×C4)⋊6Q16  Dic34Q16  Dic35Q16  Dic54Q16 ...

Matrix representation of C4×Q16 in GL4(𝔽17) generated by

4000
0400
00160
00016
,
0100
16000
001111
0030
,
01600
16000
00139
0004
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,16,0,0,0,0,16],[0,16,0,0,1,0,0,0,0,0,11,3,0,0,11,0],[0,16,0,0,16,0,0,0,0,0,13,0,0,0,9,4] >;

C4×Q16 in GAP, Magma, Sage, TeX

C_4\times Q_{16}
% in TeX

G:=Group("C4xQ16");
// GroupNames label

G:=SmallGroup(64,120);
// by ID

G=gap.SmallGroup(64,120);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,199,230,963,489,117]);
// Polycyclic

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

Export

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

׿
×
𝔽