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G = SD161D4order 128 = 27

1st semidirect product of SD16 and D4 acting via D4/C4=C2

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

Aliases: SD161D4, C42.441C23, C4.1282+ 1+4, D426C2, C83(C2×D4), C2.58D42, Q84(C2×D4), C86D45C2, C84D421C2, C82D421C2, C4⋊D836C2, C4⋊C830C22, C43(C8⋊C22), C4⋊C4.360D4, (C4×C8)⋊31C22, Q86D44C2, D4.23(C2×D4), (C2×D8)⋊7C22, D4⋊D438C2, C22⋊D829C2, C4⋊SD1619C2, (C4×SD16)⋊19C2, (C2×D4).310D4, C2.40(D4○D8), C22⋊C4.43D4, (C4×Q8)⋊20C22, C4.88(C22×D4), C4.Q852C22, C41D411C22, C4⋊C4.213C23, C4⋊D410C22, C22⋊C826C22, (C2×C8).284C23, (C2×C4).472C24, C23.314(C2×D4), D4⋊C438C22, Q8⋊C471C22, (C2×SD16)⋊29C22, (C4×D4).146C22, (C2×D4).211C23, (C2×Q8).389C23, (C2×M4(2))⋊24C22, (C22×C4).322C23, C22.732(C22×D4), (C22×D4).400C22, (C2×C8⋊C22)⋊32C2, (C2×C4).156(C2×D4), C2.73(C2×C8⋊C22), (C2×C4○D4)⋊16C22, SmallGroup(128,2006)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — SD161D4
C1C2C22C2×C4C2×D4C22×D4C2×C8⋊C22 — SD161D4
C1C2C2×C4 — SD161D4
C1C22C4×D4 — SD161D4
C1C2C2C2×C4 — SD161D4

Generators and relations for SD161D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=cac-1=a3, dad=a-1, bc=cb, bd=db, dcd=c-1 >

Subgroups: 688 in 278 conjugacy classes, 96 normal (38 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×2], C4 [×7], C22, C22 [×30], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×2], C2×C4 [×13], D4 [×2], D4 [×31], Q8 [×2], Q8, C23 [×2], C23 [×15], C42, C42, C22⋊C4 [×2], C22⋊C4 [×5], C4⋊C4 [×3], C4⋊C4, C2×C8 [×2], C2×C8 [×2], M4(2) [×4], D8 [×10], SD16 [×4], SD16 [×4], C22×C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×4], C2×D4 [×22], C2×Q8, C4○D4 [×6], C24 [×2], C4×C8, C22⋊C8 [×2], D4⋊C4, D4⋊C4 [×4], Q8⋊C4, C4⋊C8, C4.Q8, C4×D4 [×2], C4×D4, C4×Q8, C22≀C2 [×2], C4⋊D4 [×4], C4⋊D4 [×3], C41D4 [×2], C41D4, C2×M4(2) [×2], C2×D8 [×6], C2×SD16, C2×SD16 [×2], C8⋊C22 [×8], C22×D4 [×2], C22×D4, C2×C4○D4 [×2], C86D4, C4×SD16, C22⋊D8 [×2], D4⋊D4 [×2], C4⋊D8, C4⋊SD16, C82D4 [×2], C84D4, D42, Q86D4, C2×C8⋊C22 [×2], SD161D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C24, C8⋊C22 [×2], C22×D4 [×2], 2+ 1+4, D42, C2×C8⋊C22, D4○D8, SD161D4

Character table of SD161D4

 class 12A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D8E8F
 size 11114444888822224444488444488
ρ111111111111111111111111111111    trivial
ρ21111-1-1-1-111-1-111111-11-111-1-1-1-1-111    linear of order 2
ρ31111-1-1-1-111111111-1-11-1-1-1-11111-1-1    linear of order 2
ρ41111111111-1-11111-1111-1-11-1-1-1-1-1-1    linear of order 2
ρ51111-1111-11-11-111-111-1-11-1-1-1-111-11    linear of order 2
ρ611111-1-1-1-111-1-111-11-1-111-1111-1-1-11    linear of order 2
ρ711111-1-1-1-11-11-111-1-1-1-11-111-1-1111-1    linear of order 2
ρ81111-1111-111-1-111-1-11-1-1-11-111-1-11-1    linear of order 2
ρ9111111-1-1-1-1111111111111-1-1-1-1-1-1-1    linear of order 2
ρ101111-1-111-1-1-1-111111-11-11111111-1-1    linear of order 2
ρ111111-1-111-1-1111111-1-11-1-1-11-1-1-1-111    linear of order 2
ρ12111111-1-1-1-1-1-11111-1111-1-1-1111111    linear of order 2
ρ131111-11-1-11-1-11-111-111-1-11-1111-1-11-1    linear of order 2
ρ1411111-1111-11-1-111-11-1-111-1-1-1-1111-1    linear of order 2
ρ1511111-1111-1-11-111-1-1-1-11-11-111-1-1-11    linear of order 2
ρ161111-11-1-11-11-1-111-1-11-1-1-111-1-111-11    linear of order 2
ρ172222-2-20000002-2-2202-22000000000    orthogonal lifted from D4
ρ182222-22000000-2-2-2-20-222000000000    orthogonal lifted from D4
ρ192-22-200-22000002-20-200020000-2200    orthogonal lifted from D4
ρ202-22-200-22000002-202000-200002-200    orthogonal lifted from D4
ρ212222220000002-2-220-2-2-2000000000    orthogonal lifted from D4
ρ2222222-2000000-2-2-2-2022-2000000000    orthogonal lifted from D4
ρ232-22-2002-2000002-20-2000200002-200    orthogonal lifted from D4
ρ242-22-2002-2000002-202000-20000-2200    orthogonal lifted from D4
ρ254-4-4400000000400-40000000000000    orthogonal lifted from C8⋊C22
ρ264-4-4400000000-40040000000000000    orthogonal lifted from C8⋊C22
ρ274-44-4000000000-4400000000000000    orthogonal lifted from 2+ 1+4
ρ2844-4-4000000000000000000022-220000    orthogonal lifted from D4○D8
ρ2944-4-40000000000000000000-22220000    orthogonal lifted from D4○D8

Smallest permutation representation of SD161D4
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 23)(2 18)(3 21)(4 24)(5 19)(6 22)(7 17)(8 20)(9 25)(10 28)(11 31)(12 26)(13 29)(14 32)(15 27)(16 30)
(1 26 23 12)(2 29 24 15)(3 32 17 10)(4 27 18 13)(5 30 19 16)(6 25 20 11)(7 28 21 14)(8 31 22 9)
(1 12)(2 11)(3 10)(4 9)(5 16)(6 15)(7 14)(8 13)(17 32)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)

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

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

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

Matrix representation of SD161D4 in GL6(𝔽17)

1600000
0160000
0000160
000001
0001600
0016000
,
100000
010000
0016000
000100
000001
000010
,
0160000
100000
0000314
000033
00141400
0031400
,
0160000
1600000
0000314
000033
003300
0014300

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,16,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,0,14,3,0,0,0,0,14,14,0,0,3,3,0,0,0,0,14,3,0,0],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,0,3,14,0,0,0,0,3,3,0,0,3,3,0,0,0,0,14,3,0,0] >;

SD161D4 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\rtimes_1D_4
% in TeX

G:=Group("SD16:1D4");
// GroupNames label

G:=SmallGroup(128,2006);
// by ID

G=gap.SmallGroup(128,2006);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,723,346,2804,1411,375,172]);
// Polycyclic

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

Export

Character table of SD161D4 in TeX

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