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, D4○D8⋊7C2, (C2×C8)⋊2C23, D4○(C8⋊C22), D4○SD16⋊5C2, C4○D4.54D4, D4.65(C2×D4), C4○D8⋊3C22, C8○D4⋊7C22, C4○D4⋊3C23, Q8.67(C2×D4), Q8○(C8.C22), (C2×D4).337D4, (C2×D8)⋊34C22, (C2×D4)⋊12C23, Q8○M4(2)⋊3C2, (C2×Q8).256D4, (C2×Q8)⋊12C23, C2.47(D4×C23), C8⋊C22⋊15C22, C2.C25⋊6C2, (C2×C4).149C24, C4.129(C22×D4), C23.358(C2×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

Subgroups: 1220 in 732 conjugacy classes, 426 normal (13 characteristic)
C1, C2, C2 [×17], C4 [×2], C4 [×6], C4 [×6], C22, C22 [×6], C22 [×37], C8 [×8], C2×C4, C2×C4 [×15], C2×C4 [×35], D4 [×22], D4 [×49], Q8 [×10], Q8 [×7], C23 [×3], C23 [×27], C2×C8 [×12], M4(2) [×16], D8 [×24], SD16 [×32], Q16 [×8], C22×C4 [×3], C22×C4 [×9], C2×D4 [×24], C2×D4 [×54], C2×Q8 [×2], C2×Q8 [×6], C2×Q8 [×4], C4○D4 [×36], C4○D4 [×42], C24 [×3], C2×M4(2) [×6], C8○D4 [×8], C2×D8 [×12], C2×SD16 [×12], C4○D8 [×24], C8⋊C22 [×48], C8.C22 [×16], C22×D4 [×3], C22×D4 [×3], C2×C4○D4, C2×C4○D4 [×9], C2×C4○D4 [×5], 2+ (1+4) [×12], 2+ (1+4) [×7], 2- (1+4) [×4], 2- (1+4), Q8○M4(2), C2×C8⋊C22 [×6], D8⋊C22 [×6], D4○D8 [×8], D4○SD16 [×8], C2×2+ (1+4), C2.C25, D8⋊C23

Quotients:
C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C24 [×31], C22×D4 [×14], C25, D4×C23, D8⋊C23

Generators and relations
 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 >

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

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

Matrix representation ►G ⊆ 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] >;
 

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+4)C2.C25C2×D4C2×Q8C4○D4C1
# reps116688113141

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
 

׿
×
⋊
ℤ
𝔽
○
≀