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G = C8.Q16  order 128 = 27

2nd non-split extension by C8 of Q16 acting via Q16/C4=C22

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

Aliases: C32⋊1C4, C16.Q8, C8.2Q16, C4.9SD32, M6(2).1C2, C22.5SD32, (C2×C4).15D8, (C2×C8).87D4, C8.18(C4⋊C4), C16.19(C2×C4), C16⋊4C4.1C2, C2.3(C16⋊4C4), C4.13(C2.D8), C8.4Q8.3C2, (C2×C16).16C22, SmallGroup(128,158)

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

Derived series
C1 — C16 — C8.Q16
Chief series
C1 — C2 — C4 — C8 — C2×C8 — C2×C16 — M6(2) — C8.Q16
Lower central
C1 — C2 — C4 — C8 — C16 — C8.Q16
Upper central
C1 — C2 — C2×C4 — C2×C8 — C2×C16 — C8.Q16
Jennings
C1 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C4 — C4 — C4 — C4 — C8 — C8 — C2×C16 — C8.Q16

Generators and relations for C8.Q16
 G = < a,b,c | a8=1, b8=a2, c2=a-1b4, bab-1=a5, cac-1=a-1, cbc-1=a4b7 >

C1
C2
2C2
C4
C22
C4
16C4
C8
C8
C2×C4
8C2×C4
8C8
C16
C2×C8
C16
4C4⋊C4
4M4(2)
C32
C32
C2×C16
2C8.C4
2C2.D8
C8.4Q8
M6(2)
C16⋊4C4
C8.Q16

Character table of C8.Q16

 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-ii11-1-ii1111-1-111-11-11-1-1    linear of order 4
ρ611-11-1-ii11-1i-i1111-1-1-1-11-11-111    linear of order 4
ρ711-11-1i-i11-1-ii1111-1-1-1-11-11-111    linear of order 4
ρ811-11-1i-i11-1i-i1111-1-111-11-11-1-1    linear of order 4
ρ92222200-2-2-200000000-√2-√2√2√2-√2√2√2-√2    orthogonal lifted from D8
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
ρ1222-22-200-2-2200000000√2√2√2-√2-√2-√2√2-√2    symplectic lifted from Q16, Schur index 2
ρ1322-22-200-2-2200000000-√2-√2-√2√2√2√2-√2√2    symplectic lifted from Q16, Schur index 2
ρ1422-22-20022-200-2-2-2-22200000000    symplectic lifted from Q8, Schur index 2
ρ1522-2-220000000√2-√2√2-√2√2-√2ζ1615+ζ169ζ167+ζ16ζ1613+ζ1611ζ165+ζ163ζ167+ζ16ζ1613+ζ1611ζ165+ζ163ζ1615+ζ169    complex lifted from SD32
ρ1622-2-220000000√2-√2√2-√2√2-√2ζ167+ζ16ζ1615+ζ169ζ165+ζ163ζ1613+ζ1611ζ1615+ζ169ζ165+ζ163ζ1613+ζ1611ζ167+ζ16    complex lifted from SD32
ρ17222-2-20000000√2-√2√2-√2-√2√2ζ1615+ζ169ζ167+ζ16ζ165+ζ163ζ165+ζ163ζ1615+ζ169ζ1613+ζ1611ζ1613+ζ1611ζ167+ζ16    complex lifted from SD32
ρ18222-2-20000000√2-√2√2-√2-√2√2ζ167+ζ16ζ1615+ζ169ζ1613+ζ1611ζ1613+ζ1611ζ167+ζ16ζ165+ζ163ζ165+ζ163ζ1615+ζ169    complex lifted from SD32
ρ19222-2-20000000-√2√2-√2√2√2-√2ζ165+ζ163ζ1613+ζ1611ζ167+ζ16ζ167+ζ16ζ165+ζ163ζ1615+ζ169ζ1615+ζ169ζ1613+ζ1611    complex lifted from SD32
ρ2022-2-220000000-√2√2-√2√2-√2√2ζ1613+ζ1611ζ165+ζ163ζ167+ζ16ζ1615+ζ169ζ165+ζ163ζ167+ζ16ζ1615+ζ169ζ1613+ζ1611    complex lifted from SD32
ρ2122-2-220000000-√2√2-√2√2-√2√2ζ165+ζ163ζ1613+ζ1611ζ1615+ζ169ζ167+ζ16ζ1613+ζ1611ζ1615+ζ169ζ167+ζ16ζ165+ζ163    complex lifted from SD32
ρ22222-2-20000000-√2√2-√2√2√2-√2ζ1613+ζ1611ζ165+ζ163ζ1615+ζ169ζ1615+ζ169ζ1613+ζ1611ζ167+ζ16ζ167+ζ16ζ165+ζ163    complex lifted from SD32
ρ234-4000002√2-2√20002ζ165+2ζ1632ζ167+2ζ162ζ1613+2ζ16112ζ1615+2ζ1690000000000    complex faithful
ρ244-400000-2√22√20002ζ1615+2ζ1692ζ165+2ζ1632ζ167+2ζ162ζ1613+2ζ16110000000000    complex faithful
ρ254-400000-2√22√20002ζ167+2ζ162ζ1613+2ζ16112ζ1615+2ζ1692ζ165+2ζ1630000000000    complex faithful
ρ264-4000002√2-2√20002ζ1613+2ζ16112ζ1615+2ζ1692ζ165+2ζ1632ζ167+2ζ160000000000    complex faithful

Smallest permutation representation of C8.Q16
►On 32 points
Generators in S32
(1 5 9 13 17 21 25 29)(2 22 10 30 18 6 26 14)(3 7 11 15 19 23 27 31)(4 24 12 32 20 8 28 16)

(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)

(2 8 18 24)(3 15)(4 22 20 6)(5 29)(7 11)(9 25)(10 32 26 16)(12 14 28 30)(13 21)(19 31)(23 27)
 
G:=sub<Sym(32)| (1,5,9,13,17,21,25,29)(2,22,10,30,18,6,26,14)(3,7,11,15,19,23,27,31)(4,24,12,32,20,8,28,16), (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), (2,8,18,24)(3,15)(4,22,20,6)(5,29)(7,11)(9,25)(10,32,26,16)(12,14,28,30)(13,21)(19,31)(23,27)>;
 
G:=Group( (1,5,9,13,17,21,25,29)(2,22,10,30,18,6,26,14)(3,7,11,15,19,23,27,31)(4,24,12,32,20,8,28,16), (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), (2,8,18,24)(3,15)(4,22,20,6)(5,29)(7,11)(9,25)(10,32,26,16)(12,14,28,30)(13,21)(19,31)(23,27) );
 
G=PermutationGroup([[(1,5,9,13,17,21,25,29),(2,22,10,30,18,6,26,14),(3,7,11,15,19,23,27,31),(4,24,12,32,20,8,28,16)], [(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)], [(2,8,18,24),(3,15),(4,22,20,6),(5,29),(7,11),(9,25),(10,32,26,16),(12,14,28,30),(13,21),(19,31),(23,27)]])
 

Matrix representation of C8.Q16 ►in GL4(𝔽7) generated by

1651
6525
5044
4154
,
1136
2325
1315
6512
,
6066
0462
0506
0254
G:=sub<GL(4,GF(7))| [1,6,5,4,6,5,0,1,5,2,4,5,1,5,4,4],[1,2,1,6,1,3,3,5,3,2,1,1,6,5,5,2],[6,0,0,0,0,4,5,2,6,6,0,5,6,2,6,4] >;
 

C8.Q16 in GAP, Magma, Sage, TeX 

C_8.Q_{16}
 
% in TeX
 
G:=Group("C8.Q16");
 
// GroupNames label
 
G:=SmallGroup(128,158);
 
// by ID
 
G=gap.SmallGroup(128,158);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,372,422,604,1018,1684,242,4037,124]);
 
// Polycyclic
 
G:=Group<a,b,c|a^8=1,b^8=a^2,c^2=a^-1*b^4,b*a*b^-1=a^5,c*a*c^-1=a^-1,c*b*c^-1=a^4*b^7>;
 
// generators/relations
 

Export

Subgroup lattice of C8.Q16 in TeX
Character table of C8.Q16 in TeX

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