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G = D8.13D4order 128 = 27

5th non-split extension by D8 of D4 acting via D4/C22=C2

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

Aliases: D8.13D4, Q16.12D4, SD16.2D4, C42.40C23, M4(2).14C23, 2- 1+43C22, C22.22+ 1+4, 2+ 1+4.6C22, C2.73D42, Q8○D82C2, C8.36(C2×D4), C4≀C26C22, C8.26D45C2, D4○SD162C2, D4.32(C2×D4), C8○D43C22, C4⋊Q818C22, Q8.32(C2×D4), D4.5D45C2, D4.3D44C2, D4.9D46C2, D4.8D46C2, C8.2D420C2, (C2×C4).22C24, (C2×C8).93C23, (C2×D4).8C23, C8⋊C420C22, D4.10D45C2, (C2×Q8).6C23, C8.C46C22, (C2×Q16)⋊30C22, C4○D4.11C23, C4○D8.10C22, C4.103(C22×D4), C8⋊C22.1C22, C8.C222C22, C4.10D43C22, C4.D4.6C22, (C2×SD16).51C22, C4.4D4.57C22, SmallGroup(128,2021)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D8.13D4
C1C2C4C2×C4C4○D42+ 1+4D4○SD16 — D8.13D4
C1C2C2×C4 — D8.13D4
C1C2C2×C4 — D8.13D4
C1C2C2C2×C4 — D8.13D4

Generators and relations for D8.13D4
 G = < a,b,c,d | a8=b2=1, c4=d2=a4, bab=dad-1=a-1, cac-1=a3, cbc-1=a2b, dbd-1=a6b, dcd-1=c3 >

Subgroups: 444 in 224 conjugacy classes, 92 normal (18 characteristic)
C1, C2, C2 [×6], C4 [×2], C4 [×8], C22, C22 [×7], C8 [×4], C8 [×4], C2×C4, C2×C4 [×15], D4 [×4], D4 [×11], Q8 [×4], Q8 [×9], C23 [×2], C42, C22⋊C4, C4⋊C4, C2×C8 [×2], C2×C8 [×4], M4(2) [×4], M4(2) [×4], D8 [×2], D8 [×2], SD16 [×4], SD16 [×12], Q16 [×2], Q16 [×10], C2×D4, C2×D4 [×3], C2×Q8, C2×Q8 [×2], C2×Q8 [×6], C4○D4 [×4], C4○D4 [×11], C8⋊C4, C4.D4, C4.10D4, C4.10D4 [×2], C4≀C2 [×4], C8.C4 [×2], C4.4D4, C4⋊Q8, C8○D4 [×4], C2×SD16 [×2], C2×SD16 [×2], C2×Q16 [×2], C2×Q16 [×2], C4○D8 [×2], C4○D8 [×4], C8⋊C22 [×2], C8⋊C22 [×2], C8.C22 [×6], C8.C22 [×6], 2+ 1+4, 2- 1+4, 2- 1+4 [×2], C8.26D4 [×2], D4.8D4, D4.9D4, D4.10D4 [×2], D4.3D4 [×2], D4.5D4 [×2], C8.2D4, D4○SD16 [×2], Q8○D8 [×2], D8.13D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C24, C22×D4 [×2], 2+ 1+4, D42, D8.13D4

Character table of D8.13D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F8G8H
 size 11244448224444888844448888
ρ111111111111111111111111111    trivial
ρ2111111-1-11111-111-11-11-1-111-1-1-1    linear of order 2
ρ311111-111111-1111-1-1-1-111-1-1-1-11    linear of order 2
ρ411111-1-1-1111-1-1111-11-1-1-1-1-111-1    linear of order 2
ρ51111-11-1-111-11-11-1-1-111111-11-11    linear of order 2
ρ61111-111111-1111-11-1-11-1-11-1-11-1    linear of order 2
ρ71111-1-1-1-111-1-1-11-111-1-111-11-111    linear of order 2
ρ81111-1-11111-1-111-1-111-1-1-1-111-1-1    linear of order 2
ρ9111-1-1-11-111-1-11-11-11-11-1-11-1111    linear of order 2
ρ10111-1-1-1-1111-1-1-1-111111111-1-1-1-1    linear of order 2
ρ11111-1-111-111-111-111-11-1-1-1-11-1-11    linear of order 2
ρ12111-1-11-1111-11-1-11-1-1-1-111-1111-1    linear of order 2
ρ13111-11-1-11111-1-1-1-11-1-11-1-1111-11    linear of order 2
ρ14111-11-11-1111-11-1-1-1-1111111-11-1    linear of order 2
ρ15111-111-111111-1-1-1-111-1-1-1-1-1-111    linear of order 2
ρ16111-1111-111111-1-111-1-111-1-11-1-1    linear of order 2
ρ1722-2202002-20-20-20000-20020000    orthogonal lifted from D4
ρ1822-20-20-20-222020000002-200000    orthogonal lifted from D4
ρ1922-220-2002-2020-20000200-20000    orthogonal lifted from D4
ρ2022-20-2020-2220-2000000-2200000    orthogonal lifted from D4
ρ2122-2020-20-22-202000000-2200000    orthogonal lifted from D4
ρ2222-2-202002-20-2020000200-20000    orthogonal lifted from D4
ρ2322-202020-22-20-20000002-200000    orthogonal lifted from D4
ρ2422-2-20-2002-202020000-20020000    orthogonal lifted from D4
ρ2544400000-4-40000000000000000    orthogonal lifted from 2+ 1+4
ρ268-8000000000000000000000000    symplectic faithful, Schur index 2

Smallest permutation representation of D8.13D4
On 32 points
Generators in S32
(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)
(1 8)(2 7)(3 6)(4 5)(9 16)(10 15)(11 14)(12 13)(17 18)(19 24)(20 23)(21 22)(25 26)(27 32)(28 31)(29 30)
(1 30 18 13 5 26 22 9)(2 25 19 16 6 29 23 12)(3 28 20 11 7 32 24 15)(4 31 21 14 8 27 17 10)
(1 13 5 9)(2 12 6 16)(3 11 7 15)(4 10 8 14)(17 31 21 27)(18 30 22 26)(19 29 23 25)(20 28 24 32)

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

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

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

Matrix representation of D8.13D4 in GL8(𝔽17)

00000100
000016000
000000016
00000010
00010000
001600000
01000000
160000000
,
00000100
000016000
00000001
000000160
016000000
10000000
000160000
00100000
,
125000000
1212000000
005120000
00550000
000000125
0000001212
000012500
0000121200
,
125000000
55000000
005120000
0012120000
000000125
00000055
000012500
00005500

G:=sub<GL(8,GF(17))| [0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0],[12,12,0,0,0,0,0,0,5,12,0,0,0,0,0,0,0,0,5,5,0,0,0,0,0,0,12,5,0,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,5,12,0,0,0,0,12,12,0,0,0,0,0,0,5,12,0,0],[12,5,0,0,0,0,0,0,5,5,0,0,0,0,0,0,0,0,5,12,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,12,5,0,0,0,0,0,0,5,5,0,0,0,0,12,5,0,0,0,0,0,0,5,5,0,0] >;

D8.13D4 in GAP, Magma, Sage, TeX

D_8._{13}D_4
% in TeX

G:=Group("D8.13D4");
// GroupNames label

G:=SmallGroup(128,2021);
// by ID

G=gap.SmallGroup(128,2021);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,346,2804,1411,375,172,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=1,c^4=d^2=a^4,b*a*b=d*a*d^-1=a^-1,c*a*c^-1=a^3,c*b*c^-1=a^2*b,d*b*d^-1=a^6*b,d*c*d^-1=c^3>;
// generators/relations

Export

Character table of D8.13D4 in TeX

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