Copied to
clipboard

G = C4.SD16order 64 = 26

4th non-split extension by C4 of SD16 acting via SD16/C8=C2

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

Aliases: C4.3Q16, C4.4SD16, C42.77C22, (C4×C8).4C2, C4⋊Q8.7C2, (C2×C4).75D4, C2.9(C2×Q16), C4.14(C4○D4), C4⋊C4.17C22, (C2×C8).67C22, Q8⋊C4.1C2, C2.15(C2×SD16), (C2×C4).112C23, C22.108(C2×D4), (C2×Q8).20C22, C2.10(C4.4D4), SmallGroup(64,168)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C4.SD16
C1C2C4C2×C4C2×C8C4×C8 — C4.SD16
C1C2C2×C4 — C4.SD16
C1C22C42 — C4.SD16
C1C2C2C2×C4 — C4.SD16

Generators and relations for C4.SD16
 G = < a,b,c | a4=b8=1, c2=a2, ab=ba, cac-1=a-1, cbc-1=a2b3 >

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

Character table of C4.SD16

 class 12A2B2C4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F8G8H
 size 1111222222888822222222
ρ11111111111111111111111    trivial
ρ21111-1-11-11-11-11-1-1-11-11-111    linear of order 2
ρ31111111111-111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ41111-1-11-11-1-1-11111-11-11-1-1    linear of order 2
ρ511111111111-1-11-1-1-1-1-1-1-1-1    linear of order 2
ρ61111-1-11-11-111-1-111-11-11-1-1    linear of order 2
ρ71111111111-1-1-1-111111111    linear of order 2
ρ81111-1-11-11-1-11-11-1-11-11-111    linear of order 2
ρ92222-2-2-22-22000000000000    orthogonal lifted from D4
ρ10222222-2-2-2-2000000000000    orthogonal lifted from D4
ρ1122-2-200020-20000-22-222-22-2    symplectic lifted from Q16, Schur index 2
ρ1222-2-2000-2020000-2222-2-2-22    symplectic lifted from Q16, Schur index 2
ρ1322-2-200020-200002-22-2-22-22    symplectic lifted from Q16, Schur index 2
ρ1422-2-2000-20200002-2-2-2222-2    symplectic lifted from Q16, Schur index 2
ρ152-22-200-20200000002i02i0-2i-2i    complex lifted from C4○D4
ρ162-22-20020-200000-2i-2i02i02i00    complex lifted from C4○D4
ρ172-22-20020-2000002i2i0-2i0-2i00    complex lifted from C4○D4
ρ182-22-200-2020000000-2i0-2i02i2i    complex lifted from C4○D4
ρ192-2-222-200000000--2-2-2--2--2-2-2--2    complex lifted from SD16
ρ202-2-222-200000000-2--2--2-2-2--2--2-2    complex lifted from SD16
ρ212-2-22-2200000000--2-2--2--2-2-2--2-2    complex lifted from SD16
ρ222-2-22-2200000000-2--2-2-2--2--2-2--2    complex lifted from SD16

Smallest permutation representation of C4.SD16
Regular action on 64 points
Generators in S64
(1 38 15 28)(2 39 16 29)(3 40 9 30)(4 33 10 31)(5 34 11 32)(6 35 12 25)(7 36 13 26)(8 37 14 27)(17 43 51 57)(18 44 52 58)(19 45 53 59)(20 46 54 60)(21 47 55 61)(22 48 56 62)(23 41 49 63)(24 42 50 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 44 15 58)(2 61 16 47)(3 42 9 64)(4 59 10 45)(5 48 11 62)(6 57 12 43)(7 46 13 60)(8 63 14 41)(17 35 51 25)(18 28 52 38)(19 33 53 31)(20 26 54 36)(21 39 55 29)(22 32 56 34)(23 37 49 27)(24 30 50 40)

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

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

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

C4.SD16 is a maximal subgroup of
C85Q16  C8212C2  C823C2  C85SD16  C8.9SD16  C42.664C23  C42.665C23  C42.666C23  C42.667C23  C83Q16  C42.355D4  C42.241D4  C42.243D4  C42.365D4  C42.367D4  C42.259D4  C42.262D4  C42.264D4  C42.265D4  C42.267D4  C42.268D4  C42.281D4  C42.282D4  C42.283D4
 C4p.Q16: C8.7Q16  C12.14Q16  C12.9Q16  C12.Q16  C20.14Q16  C20.Q16  C20.11Q16  C28.14Q16 ...
 C4⋊C4.D2p: D4.SD16  Q8.Q16  D4.3Q16  C42.199C23  D4.5SD16  D43Q16  Q83Q16  C42.207C23 ...
C4.SD16 is a maximal quotient of
C42.55Q8  C2.(C88D4)  (C2×C4).19Q16  (C2×C4).23Q16
 C42.D2p: C42.431D4  C42.436D4  C12.14Q16  C12.9Q16  C12.Q16  C20.14Q16  C20.Q16  C20.11Q16 ...
 C4⋊C4.D2p: (C2×Q8)⋊Q8  C4⋊C4.95D4  Dic3.1Q16  Dic5.3Q16  Dic7.1Q16 ...

Matrix representation of C4.SD16 in GL4(𝔽17) generated by

01600
1000
0010
0001
,
31400
3300
0055
00125
,
7100
11000
00016
00160
G:=sub<GL(4,GF(17))| [0,1,0,0,16,0,0,0,0,0,1,0,0,0,0,1],[3,3,0,0,14,3,0,0,0,0,5,12,0,0,5,5],[7,1,0,0,1,10,0,0,0,0,0,16,0,0,16,0] >;

C4.SD16 in GAP, Magma, Sage, TeX

C_4.{\rm SD}_{16}
% in TeX

G:=Group("C4.SD16");
// GroupNames label

G:=SmallGroup(64,168);
// by ID

G=gap.SmallGroup(64,168);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,103,362,50,963,117,1444,88]);
// Polycyclic

G:=Group<a,b,c|a^4=b^8=1,c^2=a^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^2*b^3>;
// generators/relations

Export

Subgroup lattice of C4.SD16 in TeX
Character table of C4.SD16 in TeX

׿
×
𝔽