Copied to
clipboard

G = C8:11SD16order 128 = 27

2nd semidirect product of C8 and SD16 acting via SD16/D4=C2

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

Aliases: C8:11SD16, D4.1SD16, C42.222C23, C8:2C8:23C2, C4:C4.197D4, C8:3Q8:15C2, (C8xD4).13C2, (C2xC8).365D4, (C2xD4).189D4, C4.66(C4oD8), D4:2Q8.7C2, C4.57(C2xSD16), C4:Q8.45C22, C4.10D8:23C2, C2.10(C8:8D4), C4:C8.176C22, (C4xC8).251C22, C4.6Q16:14C2, D4.D4.6C2, C2.6(D4.D4), (C4xD4).277C22, C2.10(D4.3D4), C4.110(C8.C22), C22.183(C4:D4), (C2xC4).7(C4oD4), (C2xC4).1257(C2xD4), SmallGroup(128,403)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C8:11SD16
C1C2C22C2xC4C42C4xD4C8xD4 — C8:11SD16
C1C22C42 — C8:11SD16
C1C22C42 — C8:11SD16
C1C22C22C42 — C8:11SD16

Generators and relations for C8:11SD16
 G = < a,b,c | a8=b8=c2=1, bab-1=a3, ac=ca, cbc=b3 >

Subgroups: 184 in 83 conjugacy classes, 36 normal (32 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2xC4, C2xC4, D4, D4, Q8, C23, C42, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, SD16, C22xC4, C2xD4, C2xQ8, C4xC8, C22:C8, D4:C4, Q8:C4, C4:C8, C4.Q8, C4xD4, C4:Q8, C22xC8, C2xSD16, C4.10D8, C4.6Q16, C8:2C8, C8xD4, D4.D4, D4:2Q8, C8:3Q8, C8:11SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2xD4, C4oD4, C4:D4, C2xSD16, C4oD8, C8.C22, D4.D4, C8:8D4, D4.3D4, C8:11SD16

Character table of C8:11SD16

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F8G8H8I8J8K8L8M8N
 size 1111442222444161622224444448888
ρ111111111111111111111111111111    trivial
ρ21111-1-11111-11-11-1-1-1-1-1111-1-111-11-1    linear of order 2
ρ31111-1-11111-11-1-1-11111-1-1-111-11111    linear of order 2
ρ41111111111111-11-1-1-1-1-1-1-1-1-1-11-11-1    linear of order 2
ρ511111111111111-1-1-1-1-1-1-1-1-1-1-1-11-11    linear of order 2
ρ61111-1-11111-11-1111111-1-1-111-1-1-1-1-1    linear of order 2
ρ71111-1-11111-11-1-11-1-1-1-1111-1-11-11-11    linear of order 2
ρ81111111111111-1-11111111111-1-1-1-1    linear of order 2
ρ92222-2-2-22-222-220000000000000000    orthogonal lifted from D4
ρ102222002-22-20-20002222000-2-200000    orthogonal lifted from D4
ρ112222002-22-20-2000-2-2-2-20002200000    orthogonal lifted from D4
ρ12222222-22-22-2-2-20000000000000000    orthogonal lifted from D4
ρ13222200-2-2-2-2020000000-2i-2i2i002i0000    complex lifted from C4oD4
ρ14222200-2-2-2-20200000002i2i-2i00-2i0000    complex lifted from C4oD4
ρ1522-2-2000-202-2i02i00--2-2-2--2-22-2--2-220000    complex lifted from C4oD8
ρ1622-2-2000-2022i0-2i00-2--2--2-2-22-2-2--220000    complex lifted from C4oD8
ρ1722-2-22-2020-200000--2-2-2--2-2--2--2-2--2-20000    complex lifted from SD16
ρ182-22-200-2020000002-22-2000000-2-2--2--2    complex lifted from SD16
ρ1922-2-2-22020-200000--2-2-2--2--2-2-2-2--2--20000    complex lifted from SD16
ρ202-22-200-202000000-22-22000000--2-2-2--2    complex lifted from SD16
ρ212-22-200-2020000002-22-2000000--2--2-2-2    complex lifted from SD16
ρ2222-2-22-2020-200000-2--2--2-2--2-2-2--2-2--20000    complex lifted from SD16
ρ232-22-200-202000000-22-22000000-2--2--2-2    complex lifted from SD16
ρ2422-2-2000-202-2i02i00-2--2--2-22-22-2--2-20000    complex lifted from C4oD8
ρ2522-2-2000-2022i0-2i00--2-2-2--22-22--2-2-20000    complex lifted from C4oD8
ρ2622-2-2-22020-200000-2--2--2-2-2--2--2--2-2-20000    complex lifted from SD16
ρ274-44-40040-400000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ284-4-44000000000002-22-2-2-2-2-20000000000    complex lifted from D4.3D4
ρ294-4-4400000000000-2-2-2-22-22-20000000000    complex lifted from D4.3D4

Smallest permutation representation of C8:11SD16
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 12 39 20 45 32 49 58)(2 15 40 23 46 27 50 61)(3 10 33 18 47 30 51 64)(4 13 34 21 48 25 52 59)(5 16 35 24 41 28 53 62)(6 11 36 19 42 31 54 57)(7 14 37 22 43 26 55 60)(8 9 38 17 44 29 56 63)
(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 57)(32 58)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 49)(40 50)

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

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

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

Matrix representation of C8:11SD16 in GL4(F17) generated by

1000
0100
00150
00139
,
12500
121200
0072
00910
,
1000
01600
0010
001016
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,15,13,0,0,0,9],[12,12,0,0,5,12,0,0,0,0,7,9,0,0,2,10],[1,0,0,0,0,16,0,0,0,0,1,10,0,0,0,16] >;

C8:11SD16 in GAP, Magma, Sage, TeX

C_8\rtimes_{11}{\rm SD}_{16}
% in TeX

G:=Group("C8:11SD16");
// GroupNames label

G:=SmallGroup(128,403);
// by ID

G=gap.SmallGroup(128,403);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,64,422,1123,136,2804,718,172]);
// Polycyclic

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

Export

Character table of C8:11SD16 in TeX

׿
x
:
Z
F
o
wr
Q
<