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G = C2×Q16  order 32 = 25

Direct product of C2 and Q16

direct product, p-group, metabelian, nilpotent (class 3), monomial

Aliases: C2×Q16, C4.8D4, C4.3C23, C8.5C22, C22.16D4, Q8.1C22, (C2×C8).4C2, C2.13(C2×D4), (C2×Q8).4C2, (C2×C4).28C22, SmallGroup(32,41)

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

Derived series
C1 — C4 — C2×Q16
Chief series
C1 — C2 — C4 — C2×C4 — C2×Q8 — C2×Q16
Lower central
C1 — C2 — C4 — C2×Q16
Upper central
C1 — C22 — C2×C4 — C2×Q16
Jennings
C1 — C2 — C2 — C4 — C2×Q16

Generators and relations for C2×Q16
 G = < a,b,c | a2=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >

C1
C2
C2
C2
C4
C4
C22
2C4
2C4
2C4
2C4
Q8
Q8
C8
Q8
Q8
C8
C2×C4
2C2×C4
2Q8
2Q8
2C2×C4
Q16
Q16
Q16
Q16
C2×C8
C2×Q8
C2×Q8
C2×Q16

Character table of C2×Q16

 class 12A2B2C4A4B4C4D4E4F8A8B8C8D
 size 11112244442222
ρ111111111111111    trivial
ρ21-11-1-11-11-11-1-111    linear of order 2
ρ3111111-1-111-1-1-1-1    linear of order 2
ρ41-11-1-111-1-1111-1-1    linear of order 2
ρ51-11-1-11-111-111-1-1    linear of order 2
ρ611111111-1-1-1-1-1-1    linear of order 2
ρ71-11-1-111-11-1-1-111    linear of order 2
ρ8111111-1-1-1-11111    linear of order 2
ρ92222-2-200000000    orthogonal lifted from D4
ρ102-22-22-200000000    orthogonal lifted from D4
ρ1122-2-2000000-√2√2√2-√2    symplectic lifted from Q16, Schur index 2
ρ1222-2-2000000√2-√2-√2√2    symplectic lifted from Q16, Schur index 2
ρ132-2-22000000-√2√2-√2√2    symplectic lifted from Q16, Schur index 2
ρ142-2-22000000√2-√2√2-√2    symplectic lifted from Q16, Schur index 2

Smallest permutation representation of C2×Q16
►Regular action on 32 points
Generators in S32
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 25)(9 23)(10 24)(11 17)(12 18)(13 19)(14 20)(15 21)(16 22)

(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)

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

C2×Q16 is a maximal subgroup of
 C2.Q32  C8.17D4  Q16⋊C4  C22⋊Q16  D4.7D4  C4⋊2Q16  Q8.D4  C8.18D4  C8.D4  D4.5D4  C4⋊Q16  C8.12D4  C8.2D4  Q32⋊C2  Q8○D8  C3⋊S3⋊Q16
C2×Q16 is a maximal quotient of
 C22⋊Q16  C4⋊2Q16  C8.18D4  C4.Q16  C23.48D4  C4.SD16  C4⋊Q16  C8⋊2Q8  C3⋊S3⋊Q16

Matrix representation of C2×Q16 ►in GL3(𝔽17) generated by

1600
0160
0016
,
1600
006
0146
,
100
01114
016
G:=sub<GL(3,GF(17))| [16,0,0,0,16,0,0,0,16],[16,0,0,0,0,14,0,6,6],[1,0,0,0,11,1,0,14,6] >;
 

C2×Q16 in GAP, Magma, Sage, TeX 

C_2\times Q_{16}
 
% in TeX
 
G:=Group("C2xQ16");
 
// GroupNames label
 
G:=SmallGroup(32,41);
 
// by ID
 
G=gap.SmallGroup(32,41);
 
# by ID
 
G:=PCGroup([5,-2,2,2,-2,-2,80,101,86,483,248,58]);
 
// Polycyclic
 
G:=Group<a,b,c|a^2=b^8=1,c^2=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C2×Q16 in TeX
Character table of C2×Q16 in TeX

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