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G = C4.D16  order 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⋊C16⋊4C2, C8⋊1C8⋊2C2, (C2×D8).3C4, C8⋊4D4.1C2, (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

Derived series
C1 — C2×C8 — C4.D16
Chief series
C1 — C2 — C4 — C2×C4 — C42 — C4×C8 — C8⋊4D4 — C4.D16
Lower central
C1 — C2 — C2×C4 — C2×C8 — C4.D16
Upper central
C1 — C22 — C42 — C4×C8 — C4.D16
Jennings
C1 — C2 — C2 — C2 — C2 — C2×C4 — C2×C4 — C4×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 >

C1
C2
C2
C2
16C2
16C2
C4
C4
C22
C4
C4
2C4
8C22
8C22
8C22
8C22
8C22
8C22
C2×C4
C2×C4
C2×C4
C8
C8
2C8
4C23
4C23
4D4
4D4
4D4
4D4
8D4
8D4
8C8
8D4
8D4
C2×C8
C2×C8
C42
2C2×D4
2C2×D4
4D8
4C2×D4
4D8
4D8
4D8
4C16
4C2×C8
4C2×D4
C4×C8
C2×D8
C2×D8
2C2×D8
2C4⋊C8
2C4⋊1D4
2C2×C16
C8⋊4D4
C8⋊1C8
C4⋊C16
C4.D16

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
ρ112-2-2200200-20√2-√2√2-√2-√2√20000ζ167-ζ16-ζ167+ζ16-ζ167+ζ16ζ165-ζ163-ζ165+ζ163ζ167-ζ16ζ165-ζ163-ζ165+ζ163    orthogonal lifted from D16
ρ122-22-20002-2002-2-2200√2-√2-√2√200000000    orthogonal lifted from D8
ρ132222002-2-22-20000000000-√2-√2-√2√2√2-√2√2√2    orthogonal lifted from D8
ρ142-22-20002-2002-2-2200-√2√2√2-√200000000    orthogonal lifted from D8
ρ152-2-2200200-20-√2√2-√2√2√2-√20000ζ165-ζ163-ζ165+ζ163-ζ165+ζ163-ζ167+ζ16ζ167-ζ16ζ165-ζ163-ζ167+ζ16ζ167-ζ16    orthogonal lifted from D16
ρ162222002-2-22-20000000000√2√2√2-√2-√2√2-√2-√2    orthogonal lifted from D8
ρ172-2-2200200-20√2-√2√2-√2-√2√20000-ζ167+ζ16ζ167-ζ16ζ167-ζ16-ζ165+ζ163ζ165-ζ163-ζ167+ζ16-ζ165+ζ163ζ165-ζ163    orthogonal lifted from D16
ρ182-2-2200200-20-√2√2-√2√2√2-√20000-ζ165+ζ163ζ165-ζ163ζ165-ζ163ζ167-ζ16-ζ167+ζ16-ζ165+ζ163ζ167-ζ16-ζ167+ζ16    orthogonal lifted from D16
ρ192-22-20002-200-222-200-√-2√-2-√-2√-200000000    complex lifted from SD16
ρ202-2-2200-20020-√2√2-√2√2-√2√20000ζ165+ζ163ζ165+ζ163ζ1613+ζ1611ζ167+ζ16ζ167+ζ16ζ1613+ζ1611ζ1615+ζ169ζ1615+ζ169    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
ρ242-2-2200-20020√2-√2√2-√2√2-√20000ζ167+ζ16ζ167+ζ16ζ1615+ζ169ζ1613+ζ1611ζ1613+ζ1611ζ1615+ζ169ζ165+ζ163ζ165+ζ163    complex lifted from SD32
ρ252-2-2200-20020-√2√2-√2√2-√2√20000ζ1613+ζ1611ζ1613+ζ1611ζ165+ζ163ζ1615+ζ169ζ1615+ζ169ζ165+ζ163ζ167+ζ16ζ167+ζ16    complex lifted from SD32
ρ262-2-2200-20020√2-√2√2-√2√2-√20000ζ1615+ζ169ζ1615+ζ169ζ167+ζ16ζ165+ζ163ζ165+ζ163ζ167+ζ16ζ1613+ζ1611ζ1613+ζ1611    complex lifted from SD32
ρ274-44-4000-4400000000000000000000    orthogonal lifted from C4.D4
ρ2844-4-40000000-2√2-2√22√22√200000000000000    orthogonal lifted from M5(2)⋊C2
ρ2944-4-400000002√22√2-2√2-2√200000000000000    orthogonal lifted from M5(2)⋊C2

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

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

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

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