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G = C16.D4  order 128 = 27

1st non-split extension by C16 of D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C16.1D4, C23.17D8, (C2×Q32)⋊8C2, C16⋊4C4⋊5C2, (C2×C4).37D8, (C2×C8).215D4, C8.103(C2×D4), C8.48(C4○D4), C4.21(C4○D8), C2.Q32⋊17C2, C4.94(C4⋊D4), C2.22(C8⋊7D4), C2.D8.9C22, (C2×C16).24C22, (C2×C8).524C23, C22.110(C2×D8), (C22×C4).350D4, (C2×M5(2)).3C2, (C2×Q16).8C22, C8.18D4.10C2, C2.12(Q32⋊C2), (C22×C8).300C22, (C2×C4).792(C2×D4), SmallGroup(128,951)

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

Derived series
C1 — C2×C8 — C16.D4
Chief series
C1 — C2 — C4 — C2×C4 — C2×C8 — C22×C8 — C2×M5(2) — C16.D4
Lower central
C1 — C2 — C4 — C2×C8 — C16.D4
Upper central
C1 — C22 — C22×C4 — C22×C8 — C16.D4
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C16.D4

Generators and relations for C16.D4
 G = < a,b,c | a16=b4=1, c2=a8, bab-1=a7, cac-1=a-1, cbc-1=a8b-1 >

Subgroups: 168 in 75 conjugacy classes, 32 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, Q8, C23, C16, C16, C22⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C22×C4, C2×Q8, Q8⋊C4, C2.D8, C2×C16, M5(2), Q32, C22⋊Q8, C22×C8, C2×Q16, C2.Q32, C16⋊4C4, C8.18D4, C2×M5(2), C2×Q32, C16.D4
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C4○D4, C4⋊D4, C2×D8, C4○D8, C8⋊7D4, Q32⋊C2, C16.D4

Character table of C16.D4

 class 12A2B2C2D4A4B4C4D4E4F4G8A8B8C8D8E8F16A16B16C16D16E16F16G16H
 size 111142241616161622224444444444
ρ111111111111111111111111111    trivial
ρ21111111111-1-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31111-111-11-1-111111-1-1-11-11-111-1    linear of order 2
ρ41111-111-11-11-11111-1-11-11-11-1-11    linear of order 2
ρ51111-111-1-11-111111-1-11-11-11-1-11    linear of order 2
ρ61111-111-1-111-11111-1-1-11-11-111-1    linear of order 2
ρ711111111-1-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ811111111-1-1-1-111111111111111    linear of order 2
ρ9222222220000-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ102222-222-20000-2-2-2-22200000000    orthogonal lifted from D4
ρ112-22-20-2200000-222-20020-20-2002    orthogonal lifted from D4
ρ122-22-20-2200000-222-200-2020200-2    orthogonal lifted from D4
ρ132222-2-2-220000000000√2-√2√2-√2-√2√2√2-√2    orthogonal lifted from D8
ρ142222-2-2-220000000000-√2√2-√2√2√2-√2-√2√2    orthogonal lifted from D8
ρ1522222-2-2-20000000000-√2-√2-√2-√2√2√2√2√2    orthogonal lifted from D8
ρ1622222-2-2-20000000000√2√2√2√2-√2-√2-√2-√2    orthogonal lifted from D8
ρ172-22-202-2000000000-2i2i√2-√-2-√2√-2√2-√-2√-2-√2    complex lifted from C4○D8
ρ182-22-20-22000002-2-220002i0-2i0-2i2i0    complex lifted from C4○D4
ρ192-22-20-22000002-2-22000-2i02i02i-2i0    complex lifted from C4○D4
ρ202-22-202-20000000002i-2i√2√-2-√2-√-2√2√-2-√-2-√2    complex lifted from C4○D8
ρ212-22-202-20000000002i-2i-√2-√-2√2√-2-√2-√-2√-2√2    complex lifted from C4○D8
ρ222-22-202-2000000000-2i2i-√2√-2√2-√-2-√2√-2-√-2√2    complex lifted from C4○D8
ρ234-4-4400000000-2√22√2-2√22√20000000000    symplectic lifted from Q32⋊C2, Schur index 2
ρ2444-4-400000000-2√2-2√22√22√20000000000    symplectic lifted from Q32⋊C2, Schur index 2
ρ254-4-44000000002√2-2√22√2-2√20000000000    symplectic lifted from Q32⋊C2, Schur index 2
ρ2644-4-4000000002√22√2-2√2-2√20000000000    symplectic lifted from Q32⋊C2, Schur index 2

Smallest permutation representation of C16.D4
►On 64 points
Generators in S64
(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)(33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64)

(1 18 39 63)(2 25 40 54)(3 32 41 61)(4 23 42 52)(5 30 43 59)(6 21 44 50)(7 28 45 57)(8 19 46 64)(9 26 47 55)(10 17 48 62)(11 24 33 53)(12 31 34 60)(13 22 35 51)(14 29 36 58)(15 20 37 49)(16 27 38 56)

(1 55 9 63)(2 54 10 62)(3 53 11 61)(4 52 12 60)(5 51 13 59)(6 50 14 58)(7 49 15 57)(8 64 16 56)(17 40 25 48)(18 39 26 47)(19 38 27 46)(20 37 28 45)(21 36 29 44)(22 35 30 43)(23 34 31 42)(24 33 32 41)
 
G:=sub<Sym(64)| (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64), (1,18,39,63)(2,25,40,54)(3,32,41,61)(4,23,42,52)(5,30,43,59)(6,21,44,50)(7,28,45,57)(8,19,46,64)(9,26,47,55)(10,17,48,62)(11,24,33,53)(12,31,34,60)(13,22,35,51)(14,29,36,58)(15,20,37,49)(16,27,38,56), (1,55,9,63)(2,54,10,62)(3,53,11,61)(4,52,12,60)(5,51,13,59)(6,50,14,58)(7,49,15,57)(8,64,16,56)(17,40,25,48)(18,39,26,47)(19,38,27,46)(20,37,28,45)(21,36,29,44)(22,35,30,43)(23,34,31,42)(24,33,32,41)>;
 
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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64), (1,18,39,63)(2,25,40,54)(3,32,41,61)(4,23,42,52)(5,30,43,59)(6,21,44,50)(7,28,45,57)(8,19,46,64)(9,26,47,55)(10,17,48,62)(11,24,33,53)(12,31,34,60)(13,22,35,51)(14,29,36,58)(15,20,37,49)(16,27,38,56), (1,55,9,63)(2,54,10,62)(3,53,11,61)(4,52,12,60)(5,51,13,59)(6,50,14,58)(7,49,15,57)(8,64,16,56)(17,40,25,48)(18,39,26,47)(19,38,27,46)(20,37,28,45)(21,36,29,44)(22,35,30,43)(23,34,31,42)(24,33,32,41) );
 
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),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)], [(1,18,39,63),(2,25,40,54),(3,32,41,61),(4,23,42,52),(5,30,43,59),(6,21,44,50),(7,28,45,57),(8,19,46,64),(9,26,47,55),(10,17,48,62),(11,24,33,53),(12,31,34,60),(13,22,35,51),(14,29,36,58),(15,20,37,49),(16,27,38,56)], [(1,55,9,63),(2,54,10,62),(3,53,11,61),(4,52,12,60),(5,51,13,59),(6,50,14,58),(7,49,15,57),(8,64,16,56),(17,40,25,48),(18,39,26,47),(19,38,27,46),(20,37,28,45),(21,36,29,44),(22,35,30,43),(23,34,31,42),(24,33,32,41)]])
 

Matrix representation of C16.D4 ►in GL6(𝔽17)

1600000
0160000
001612150
0003015
002515
0076014
,
720000
9100000
0056107
0012777
001616313
00121322
,
720000
10100000
0056107
0012777
00411313
007822

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,2,7,0,0,12,3,5,6,0,0,15,0,1,0,0,0,0,15,5,14],[7,9,0,0,0,0,2,10,0,0,0,0,0,0,5,12,16,12,0,0,6,7,16,13,0,0,10,7,3,2,0,0,7,7,13,2],[7,10,0,0,0,0,2,10,0,0,0,0,0,0,5,12,4,7,0,0,6,7,11,8,0,0,10,7,3,2,0,0,7,7,13,2] >;
 

C16.D4 in GAP, Magma, Sage, TeX 

C_{16}.D_4
 
% in TeX
 
G:=Group("C16.D4");
 
// GroupNames label
 
G:=SmallGroup(128,951);
 
// by ID
 
G=gap.SmallGroup(128,951);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,448,141,288,422,723,1123,360,4037,124]);
 
// Polycyclic
 
G:=Group<a,b,c|a^16=b^4=1,c^2=a^8,b*a*b^-1=a^7,c*a*c^-1=a^-1,c*b*c^-1=a^8*b^-1>;
 
// generators/relations
 

Export

Character table of C16.D4 in TeX

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