Copied to
clipboard

G = D8.C8  order 128 = 27

The non-split extension by D8 of C8 acting via C8/C4=C2

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

Aliases: D8.C8, Q16.C8, SD16.C8, C16.26D4, M5(2).25C22, C4≀C2.C4, C8.6(C2×C8), D4○C16⋊7C2, D4.C8⋊6C2, C4○D8.6C4, C8○D8.3C2, D4.4(C2×C8), C2.19(C8×D4), Q8.4(C2×C8), C16⋊5C4⋊11C2, C4.179(C4×D4), C8.142(C2×D4), C8.C8⋊13C2, C8.C4.6C4, C8.61(C4○D4), C4.16(C22×C8), (C2×C16).56C22, C42.172(C2×C4), (C2×C8).608C23, (C4×C8).160C22, C8○D4.18C22, C22.2(C8○D4), M4(2).24(C2×C4), (C2×C8).93(C2×C4), C4○D4.19(C2×C4), (C2×C4).447(C22×C4), SmallGroup(128,903)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C4 — D8.C8
Chief series
C1 — C2 — C4 — C8 — C2×C8 — C2×C16 — D4○C16 — D8.C8
Lower central
C1 — C2 — C4 — D8.C8
Upper central
C1 — C8 — C2×C16 — D8.C8
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — D8.C8

Generators and relations for D8.C8
 G = < a,b,c | a8=b2=1, c8=a4, bab=a-1, cac-1=a3, cbc-1=a4b >

Subgroups: 104 in 72 conjugacy classes, 46 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, C16, C16, C42, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C4○D4, C4×C8, C4≀C2, C8.C4, C2×C16, C2×C16, M5(2), M5(2), C8○D4, C4○D8, C16⋊5C4, D4.C8, C8.C8, C8○D8, D4○C16, D8.C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C2×C8, C22×C4, C2×D4, C4○D4, C4×D4, C22×C8, C8○D4, C8×D4, D8.C8

Smallest permutation representation of D8.C8
►On 32 points
Generators in S32
(1 11 5 15 9 3 13 7)(2 16 14 12 10 8 6 4)(17 23 29 19 25 31 21 27)(18 20 22 24 26 28 30 32)

(1 27)(2 20)(3 29)(4 22)(5 31)(6 24)(7 17)(8 26)(9 19)(10 28)(11 21)(12 30)(13 23)(14 32)(15 25)(16 18)

(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)
 
G:=sub<Sym(32)| (1,11,5,15,9,3,13,7)(2,16,14,12,10,8,6,4)(17,23,29,19,25,31,21,27)(18,20,22,24,26,28,30,32), (1,27)(2,20)(3,29)(4,22)(5,31)(6,24)(7,17)(8,26)(9,19)(10,28)(11,21)(12,30)(13,23)(14,32)(15,25)(16,18), (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)>;
 
G:=Group( (1,11,5,15,9,3,13,7)(2,16,14,12,10,8,6,4)(17,23,29,19,25,31,21,27)(18,20,22,24,26,28,30,32), (1,27)(2,20)(3,29)(4,22)(5,31)(6,24)(7,17)(8,26)(9,19)(10,28)(11,21)(12,30)(13,23)(14,32)(15,25)(16,18), (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) );
 
G=PermutationGroup([[(1,11,5,15,9,3,13,7),(2,16,14,12,10,8,6,4),(17,23,29,19,25,31,21,27),(18,20,22,24,26,28,30,32)], [(1,27),(2,20),(3,29),(4,22),(5,31),(6,24),(7,17),(8,26),(9,19),(10,28),(11,21),(12,30),(13,23),(14,32),(15,25),(16,18)], [(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)]])
 

44 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G8A8B8C8D8E8F8G···8L16A···16H16I···16T
order1222244444448888888···816···1616···16
size1124411244441111224···42···24···4

44 irreducible representations

dim1111111111112224
type+++++++
imageC1C2C2C2C2C2C4C4C4C8C8C8D4C4○D4C8○D4D8.C8
kernelD8.C8C16⋊5C4D4.C8C8.C8C8○D8D4○C16C4≀C2C8.C4C4○D8D8SD16Q16C16C8C22C1
# reps1121124224842244

Matrix representation of D8.C8 ►in GL4(𝔽17) generated by

2000
01500
0090
0008
,
0090
0008
2000
01500
,
0008
0010
01500
4000
G:=sub<GL(4,GF(17))| [2,0,0,0,0,15,0,0,0,0,9,0,0,0,0,8],[0,0,2,0,0,0,0,15,9,0,0,0,0,8,0,0],[0,0,0,4,0,0,15,0,0,1,0,0,8,0,0,0] >;
 

D8.C8 in GAP, Magma, Sage, TeX 

D_8.C_8
 
% in TeX
 
G:=Group("D8.C8");
 
// GroupNames label
 
G:=SmallGroup(128,903);
 
// by ID
 
G=gap.SmallGroup(128,903);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,1430,100,2019,1018,248,102,124]);
 
// Polycyclic
 
G:=Group<a,b,c|a^8=b^2=1,c^8=a^4,b*a*b=a^-1,c*a*c^-1=a^3,c*b*c^-1=a^4*b>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁