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G = C16.18D4  order 128 = 27

4th non-split extension by C16 of D4 acting via D4/C22=C2

p-group, metabelian, nilpotent (class 5), monomial

Aliases: C16.18D4, C4.12D16, Q32.2C4, C8.6SD16, M6(2).3C2, C22.4SD32, C16.6(C2×C4), (C2×C8).86D4, (C2×C4).14D8, (C2×Q32).5C2, C8.4Q8.2C2, C8.18(C22⋊C4), (C2×C16).15C22, C4.18(D4⋊C4), C2.11(C2.D16), SmallGroup(128,152)

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

Derived series
C1 — C16 — C16.18D4
Chief series
C1 — C2 — C4 — C8 — C2×C8 — C2×C16 — C2×Q32 — C16.18D4
Lower central
C1 — C2 — C4 — C8 — C16 — C16.18D4
Upper central
C1 — C2 — C2×C4 — C2×C8 — C2×C16 — C16.18D4
Jennings
C1 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C4 — C4 — C4 — C4 — C8 — C8 — C2×C16 — C16.18D4

Generators and relations for C16.18D4
 G = < a,b,c | a16=1, b4=a8, c2=bab-1=a-1, ac=ca, cbc-1=a-1b3 >

C1
C2
2C2
C4
C22
C4
8C4
8C4
C2×C4
C8
C8
4Q8
4Q8
8Q8
8C2×C4
8C8
C2×C8
C16
C16
2Q16
2Q16
4C2×Q8
4Q16
4M4(2)
Q32
C2×C16
Q32
2C32
2C8.C4
2Q32
2C2×Q16
C2×Q32
C8.4Q8
M6(2)
C16.18D4

Character table of C16.18D4

 class 12A2B4A4B4C4D8A8B8C8D8E16A16B16C16D16E16F32A32B32C32D32E32F32G32H
 size 112221616224161622224444444444
ρ111111111111111111111111111    trivial
ρ21111111111-1-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ311111-1-111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ411111-1-1111-1-111111111111111    linear of order 2
ρ511-11-1-1111-1-ii-1-1-1-111-ii-ii-iii-i    linear of order 4
ρ611-11-1-1111-1i-i-1-1-1-111i-ii-ii-i-ii    linear of order 4
ρ711-11-11-111-1-ii-1-1-1-111i-ii-ii-i-ii    linear of order 4
ρ811-11-11-111-1i-i-1-1-1-111-ii-ii-iii-i    linear of order 4
ρ922-22-20022-2002222-2-200000000    orthogonal lifted from D4
ρ10222220022200-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ112222200-2-2-200000000√2√2-√2-√2√2-√2√2-√2    orthogonal lifted from D8
ρ122222200-2-2-200000000-√2-√2√2√2-√2√2-√2√2    orthogonal lifted from D8
ρ1322-2-220000000√2-√2√2-√2-√2√2ζ165-ζ163ζ165-ζ163-ζ1615+ζ169-ζ1615+ζ169-ζ165+ζ163ζ1615-ζ169-ζ165+ζ163ζ1615-ζ169    orthogonal lifted from D16
ρ1422-2-220000000√2-√2√2-√2-√2√2-ζ165+ζ163-ζ165+ζ163ζ1615-ζ169ζ1615-ζ169ζ165-ζ163-ζ1615+ζ169ζ165-ζ163-ζ1615+ζ169    orthogonal lifted from D16
ρ1522-2-220000000-√2√2-√2√2√2-√2ζ1615-ζ169ζ1615-ζ169ζ165-ζ163ζ165-ζ163-ζ1615+ζ169-ζ165+ζ163-ζ1615+ζ169-ζ165+ζ163    orthogonal lifted from D16
ρ1622-2-220000000-√2√2-√2√2√2-√2-ζ1615+ζ169-ζ1615+ζ169-ζ165+ζ163-ζ165+ζ163ζ1615-ζ169ζ165-ζ163ζ1615-ζ169ζ165-ζ163    orthogonal lifted from D16
ρ1722-22-200-2-2200000000√-2-√-2-√-2√-2√-2√-2-√-2-√-2    complex lifted from SD16
ρ1822-22-200-2-2200000000-√-2√-2√-2-√-2-√-2-√-2√-2√-2    complex lifted from SD16
ρ19222-2-20000000-√2√2-√2√2-√2√2ζ167+ζ16ζ1615+ζ169ζ165+ζ163ζ1613+ζ1611ζ1615+ζ169ζ165+ζ163ζ167+ζ16ζ1613+ζ1611    complex lifted from SD32
ρ20222-2-20000000-√2√2-√2√2-√2√2ζ1615+ζ169ζ167+ζ16ζ1613+ζ1611ζ165+ζ163ζ167+ζ16ζ1613+ζ1611ζ1615+ζ169ζ165+ζ163    complex lifted from SD32
ρ21222-2-20000000√2-√2√2-√2√2-√2ζ165+ζ163ζ1613+ζ1611ζ1615+ζ169ζ167+ζ16ζ1613+ζ1611ζ1615+ζ169ζ165+ζ163ζ167+ζ16    complex lifted from SD32
ρ22222-2-20000000√2-√2√2-√2√2-√2ζ1613+ζ1611ζ165+ζ163ζ167+ζ16ζ1615+ζ169ζ165+ζ163ζ167+ζ16ζ1613+ζ1611ζ1615+ζ169    complex lifted from SD32
ρ234-400000-2√22√2000-2ζ165+2ζ163-2ζ1615+2ζ1692ζ165-2ζ1632ζ1615-2ζ1690000000000    symplectic faithful, Schur index 2
ρ244-4000002√2-2√20002ζ1615-2ζ169-2ζ165+2ζ163-2ζ1615+2ζ1692ζ165-2ζ1630000000000    symplectic faithful, Schur index 2
ρ254-400000-2√22√20002ζ165-2ζ1632ζ1615-2ζ169-2ζ165+2ζ163-2ζ1615+2ζ1690000000000    symplectic faithful, Schur index 2
ρ264-4000002√2-2√2000-2ζ1615+2ζ1692ζ165-2ζ1632ζ1615-2ζ169-2ζ165+2ζ1630000000000    symplectic faithful, Schur index 2

Smallest permutation representation of C16.18D4
►On 64 points
Generators in S64
(1 31 29 27 25 23 21 19 17 15 13 11 9 7 5 3)(2 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4)(33 63 61 59 57 55 53 51 49 47 45 43 41 39 37 35)(34 64 62 60 58 56 54 52 50 48 46 44 42 40 38 36)

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

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

Matrix representation of C16.18D4 ►in GL4(𝔽97) generated by

22600
71200
00271
00262
,
0010
0001
09600
1000
,
00022
00220
105300
538700
G:=sub<GL(4,GF(97))| [2,71,0,0,26,2,0,0,0,0,2,26,0,0,71,2],[0,0,0,1,0,0,96,0,1,0,0,0,0,1,0,0],[0,0,10,53,0,0,53,87,0,22,0,0,22,0,0,0] >;
 

C16.18D4 in GAP, Magma, Sage, TeX 

C_{16}._{18}D_4
 
% in TeX
 
G:=Group("C16.18D4");
 
// GroupNames label
 
G:=SmallGroup(128,152);
 
// by ID
 
G=gap.SmallGroup(128,152);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,456,422,891,604,1018,248,1684,851,242,4037,2028,124]);
 
// Polycyclic
 
G:=Group<a,b,c|a^16=1,b^4=a^8,c^2=b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b^3>;
 
// generators/relations
 

Export

Subgroup lattice of C16.18D4 in TeX
Character table of C16.18D4 in TeX

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