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G = D8⋊C23order 128 = 27

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

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

Aliases: D86C23, C8.7C24, C4.12C25, Q166C23, D4.9C24, Q8.9C24, SD165C23, M4(2)⋊9C23, 2+ 1+411C22, 2- 1+410C22, D4○D87C2, (C2×C8)⋊2C23, D4(C8⋊C22), D4○SD165C2, C4○D4.54D4, D4.65(C2×D4), C4○D43C23, C8○D47C22, C4○D83C22, 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⋊C2215C22, C2.C256C2, (C2×C4).149C24, C23.358(C2×D4), C4.129(C22×D4), D8⋊C2211C2, (C2×SD16)⋊36C22, (C22×D4)⋊51C22, C8.C2218C22, 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

C1C4 — D8⋊C23
C1C2C4C2×C4C22×C4C2×C4○D4C2×2+ 1+4 — D8⋊C23
C1C2C4 — D8⋊C23
C1C2C2×C4○D4 — D8⋊C23
C1C2C2C4 — 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 [×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

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