Copied to
clipboard

G = D8:C23  order 128 = 27

6th semidirect product of D8 and C23 acting via C23/C22=C2

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

Aliases: D8:6C23, C8.7C24, C4.12C25, Q16:6C23, D4.9C24, Q8.9C24, SD16:5C23, M4(2):9C23, 2+ 1+4:11C22, 2- 1+4:10C22, D4oD8:7C2, (C2xC8):2C23, D4o(C8:C22), D4oSD16:5C2, C4oD4.54D4, D4.65(C2xD4), C4oD4:3C23, C8oD4:7C22, C4oD8:3C22, Q8.67(C2xD4), Q8o(C8.C22), (C2xD4).337D4, (C2xD4):12C23, (C2xD8):34C22, Q8oM4(2):3C2, (C2xQ8).256D4, (C2xQ8):12C23, C2.47(D4xC23), C8:C22:15C22, C2.C25:6C2, (C2xC4).149C24, C23.358(C2xD4), C4.129(C22xD4), D8:C22:11C2, (C2xSD16):36C22, (C22xD4):51C22, C8.C22:18C22, C22.21(C22xD4), (C2xM4(2)):34C22, (C22xC4).417C23, (C2x2+ 1+4):15C2, (C2xC8:C22):36C2, (C2xC4).670(C2xD4), (C2xC4oD4):59C22, SmallGroup(128,2317)

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

Derived series
C1 — C4 — D8:C23
Chief series
C1 — C2 — C4 — C2xC4 — C22xC4 — C2xC4oD4 — C2x2+ 1+4 — D8:C23
Lower central
C1 — C2 — C4 — D8:C23
Upper central
C1 — C2 — C2xC4oD4 — D8:C23
Jennings
C1 — C2 — C2 — C4 — D8:C23

Generators and relations for D8:C23
 G = < a,b,c,d,e | a8=b2=c2=d2=e2=1, bab=a-1, cac=dad=a3, eae=a5, cbc=a2b, dbd=a6b, ebe=a4b, cd=dc, ce=ec, de=ed >

Subgroups: 1220 in 732 conjugacy classes, 426 normal (13 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C23, C2xC8, M4(2), D8, SD16, Q16, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C2xQ8, C2xQ8, C4oD4, C4oD4, C24, C2xM4(2), C8oD4, C2xD8, C2xSD16, C4oD8, C8:C22, C8.C22, C22xD4, C22xD4, C2xC4oD4, C2xC4oD4, C2xC4oD4, 2+ 1+4, 2+ 1+4, 2- 1+4, 2- 1+4, Q8oM4(2), C2xC8:C22, D8:C22, D4oD8, D4oSD16, C2x2+ 1+4, C2.C25, D8:C23
Quotients: C1, C2, C22, D4, C23, C2xD4, C24, C22xD4, C25, D4xC23, D8:C23

Permutation representations of D8:C23
►On 16 points - transitive group 16T204
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)

(1 7)(3 5)(4 8)(9 11)(10 14)(13 15)

(1 7)(3 5)(4 8)(9 15)(11 13)(12 16)

(1 5)(3 7)(9 13)(11 15)
 
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9), (1,7)(3,5)(4,8)(9,11)(10,14)(13,15), (1,7)(3,5)(4,8)(9,15)(11,13)(12,16), (1,5)(3,7)(9,13)(11,15)>;
 
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9), (1,7)(3,5)(4,8)(9,11)(10,14)(13,15), (1,7)(3,5)(4,8)(9,15)(11,13)(12,16), (1,5)(3,7)(9,13)(11,15) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)], [(1,7),(3,5),(4,8),(9,11),(10,14),(13,15)], [(1,7),(3,5),(4,8),(9,15),(11,13),(12,16)], [(1,5),(3,7),(9,13),(11,15)]])
 
G:=TransitiveGroup(16,204);
 

41 conjugacy classes

class 1 2A2B···2H2I···2R4A···4H4I···4N8A···8H
order122···22···24···44···48···8
size112···24···42···24···44···4

41 irreducible representations

dim111111112228
type++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D8:C23
kernelD8:C23Q8oM4(2)C2xC8:C22D8:C22D4oD8D4oSD16C2x2+ 1+4C2.C25C2xD4C2xQ8C4oD4C1
# reps116688113141

Matrix representation of D8:C23 ►in GL8(Z)

00000001
00000010
00000-100
0000-1000
00-100000
00010000
10000000
0-1000000
,
00000-100
00001000
00000001
000000-10
01000000
-10000000
000-10000
00100000
,
0-1000000
-10000000
00010000
00100000
0000-1000
00000100
00000010
0000000-1
,
0-1000000
-10000000
000-10000
00-100000
00001000
00000-100
00000010
0000000-1
,
10000000
01000000
00100000
00010000
0000-1000
00000-100
000000-10
0000000-1

G:=sub<GL(8,Integers())| [0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0],[0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1],[0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1] >;
 

D8:C23 in GAP, Magma, Sage, TeX 

D_8\rtimes C_2^3
 
% in TeX
 
G:=Group("D8:C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,2317);
 
// by ID
 
G=gap.SmallGroup(128,2317);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,-2,477,521,2804,4037,2028,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^8=b^2=c^2=d^2=e^2=1,b*a*b=a^-1,c*a*c=d*a*d=a^3,e*a*e=a^5,c*b*c=a^2*b,d*b*d=a^6*b,e*b*e=a^4*b,c*d=d*c,c*e=e*c,d*e=e*d>;
 
// generators/relations
 

׿
x
:
Z
F
o
wr
Q
<