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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, D4○D8⋊7C2, (C2×C8)⋊2C23, D4○(C8⋊C22), D4○SD16⋊5C2, C4○D4.54D4, D4.65(C2×D4), C4○D4⋊3C23, C8○D4⋊7C22, C4○D8⋊3C22, Q8.67(C2×D4), Q8○(C8.C22), (C2×D4).337D4, (C2×D4)⋊12C23, (C2×D8)⋊34C22, Q8○M4(2)⋊3C2, (C2×Q8).256D4, (C2×Q8)⋊12C23, C2.47(D4×C23), C8⋊C22⋊15C22, C2.C25⋊6C2, (C2×C4).149C24, C23.358(C2×D4), C4.129(C22×D4), D8⋊C22⋊11C2, (C2×SD16)⋊36C22, (C22×D4)⋊51C22, C8.C22⋊18C22, C22.21(C22×D4), (C2×M4(2))⋊34C22, (C22×C4).417C23, (C2×2+ 1+4)⋊15C2, (C2×C8⋊C22)⋊36C2, (C2×C4).670(C2×D4), (C2×C4○D4)⋊59C22, SmallGroup(128,2317)

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

Derived series
C1 — C4 — D8⋊C23
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C2×2+ 1+4 — D8⋊C23
Lower central
C1 — C2 — C4 — D8⋊C23
Upper central
C1 — C2 — C2×C4○D4 — 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, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4○D4, C24, C2×M4(2), C8○D4, C2×D8, C2×SD16, C4○D8, C8⋊C22, C8.C22, C22×D4, C22×D4, C2×C4○D4, C2×C4○D4, C2×C4○D4, 2+ 1+4, 2+ 1+4, 2- 1+4, 2- 1+4, Q8○M4(2), C2×C8⋊C22, D8⋊C22, D4○D8, D4○SD16, C2×2+ 1+4, C2.C25, D8⋊C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, C25, D4×C23, 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⋊C23Q8○M4(2)C2×C8⋊C22D8⋊C22D4○D8D4○SD16C2×2+ 1+4C2.C25C2×D4C2×Q8C4○D4C1
# 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
 

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