Copied to
clipboard

G = C4.D16order 128 = 27

1st non-split extension by C4 of D16 acting via D16/D8=C2

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

Aliases: C4.9D16, C8.26D8, C4.11SD32, C8.32SD16, C42.36D4, C4⋊C164C2, C81C82C2, (C2×D8).3C4, C84D4.1C2, (C2×C8).333D4, (C2×C4).119D8, (C4×C8).33C22, (C2×C4).16SD16, C4.5(D4⋊C4), C2.4(C2.D16), C2.7(C4.D8), C4.2(C4.D4), C2.4(M5(2)⋊C2), C22.60(D4⋊C4), (C2×C8).22(C2×C4), (C2×C4).222(C22⋊C4), SmallGroup(128,93)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — C4.D16
C1C2C4C2×C4C42C4×C8C84D4 — C4.D16
C1C2C2×C4C2×C8 — C4.D16
C1C22C42C4×C8 — C4.D16
C1C2C2C2C2C2×C4C2×C4C4×C8 — C4.D16

Generators and relations for C4.D16
 G = < a,b,c | a4=b16=1, c2=a, bab-1=a-1, ac=ca, cbc-1=ab-1 >

16C2
16C2
2C4
8C22
8C22
8C22
8C22
8C22
8C22
2C8
4C23
4C23
4D4
4D4
4D4
4D4
8D4
8D4
8C8
8D4
8D4
2C2×D4
2C2×D4
4D8
4C2×D4
4D8
4D8
4D8
4C16
4C2×C8
4C2×D4
2C2×D8
2C4⋊C8
2C41D4
2C2×C16

Character table of C4.D16

 class 12A2B2C2D2E4A4B4C4D4E8A8B8C8D8E8F8G8H8I8J16A16B16C16D16E16F16G16H
 size 1111161622224222244888844444444
ρ111111111111111111111111111111    trivial
ρ21111-1-111111111111-1-1-1-111111111    linear of order 2
ρ311111111111111111-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ41111-1-1111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ51111-11-111-1-1-1-1-1-111-i-iii-ii-i-iiii-i    linear of order 4
ρ61111-11-111-1-1-1-1-1-111ii-i-ii-iii-i-i-ii    linear of order 4
ρ711111-1-111-1-1-1-1-1-111ii-i-i-ii-i-iiii-i    linear of order 4
ρ811111-1-111-1-1-1-1-1-111-i-iiii-iii-i-i-ii    linear of order 4
ρ9222200-222-2-22222-2-2000000000000    orthogonal lifted from D4
ρ1022220022222-2-2-2-2-2-2000000000000    orthogonal lifted from D4
ρ112-2-2200200-202-22-2-220000ζ167161671616716ζ165163165163ζ16716ζ165163165163    orthogonal lifted from D16
ρ122-22-20002-2002-2-22002-2-2200000000    orthogonal lifted from D8
ρ132222002-2-22-20000000000-2-2-222-222    orthogonal lifted from D8
ρ142-22-20002-2002-2-2200-222-200000000    orthogonal lifted from D8
ρ152-2-2200200-20-22-222-20000ζ16516316516316516316716ζ16716ζ16516316716ζ16716    orthogonal lifted from D16
ρ162222002-2-22-20000000000222-2-22-2-2    orthogonal lifted from D8
ρ172-2-2200200-202-22-2-22000016716ζ16716ζ16716165163ζ16516316716165163ζ165163    orthogonal lifted from D16
ρ182-2-2200200-20-22-222-20000165163ζ165163ζ165163ζ1671616716165163ζ1671616716    orthogonal lifted from D16
ρ192-22-20002-200-222-200--2-2--2-200000000    complex lifted from SD16
ρ202-2-2200-20020-22-22-220000ζ165163ζ165163ζ16131611ζ16716ζ16716ζ16131611ζ1615169ζ1615169    complex lifted from SD32
ρ212-22-20002-200-222-200-2--2-2--200000000    complex lifted from SD16
ρ22222200-2-2-2-220000000000-2--2-2--2-2--2-2--2    complex lifted from SD16
ρ23222200-2-2-2-220000000000--2-2--2-2--2-2--2-2    complex lifted from SD16
ρ242-2-2200-200202-22-22-20000ζ16716ζ16716ζ1615169ζ16131611ζ16131611ζ1615169ζ165163ζ165163    complex lifted from SD32
ρ252-2-2200-20020-22-22-220000ζ16131611ζ16131611ζ165163ζ1615169ζ1615169ζ165163ζ16716ζ16716    complex lifted from SD32
ρ262-2-2200-200202-22-22-20000ζ1615169ζ1615169ζ16716ζ165163ζ165163ζ16716ζ16131611ζ16131611    complex lifted from SD32
ρ274-44-4000-4400000000000000000000    orthogonal lifted from C4.D4
ρ2844-4-40000000-22-22222200000000000000    orthogonal lifted from M5(2)⋊C2
ρ2944-4-400000002222-22-2200000000000000    orthogonal lifted from M5(2)⋊C2

Smallest permutation representation of C4.D16
On 64 points
Generators in S64
(1 17 45 52)(2 53 46 18)(3 19 47 54)(4 55 48 20)(5 21 33 56)(6 57 34 22)(7 23 35 58)(8 59 36 24)(9 25 37 60)(10 61 38 26)(11 27 39 62)(12 63 40 28)(13 29 41 64)(14 49 42 30)(15 31 43 50)(16 51 44 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)(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 51 17 44 45 32 52 16)(2 15 53 31 46 43 18 50)(3 49 19 42 47 30 54 14)(4 13 55 29 48 41 20 64)(5 63 21 40 33 28 56 12)(6 11 57 27 34 39 22 62)(7 61 23 38 35 26 58 10)(8 9 59 25 36 37 24 60)

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

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

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

Matrix representation of C4.D16 in GL4(𝔽17) generated by

0100
16000
0010
0001
,
12500
5500
00108
00132
,
51200
5500
00108
00117
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[12,5,0,0,5,5,0,0,0,0,10,13,0,0,8,2],[5,5,0,0,12,5,0,0,0,0,10,11,0,0,8,7] >;

C4.D16 in GAP, Magma, Sage, TeX

C_4.D_{16}
% in TeX

G:=Group("C4.D16");
// GroupNames label

G:=SmallGroup(128,93);
// by ID

G=gap.SmallGroup(128,93);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,422,387,520,1690,416,2804,1411,172,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c|a^4=b^16=1,c^2=a,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a*b^-1>;
// generators/relations

Export

Subgroup lattice of C4.D16 in TeX
Character table of C4.D16 in TeX

׿
×
𝔽