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G = C2xQ16  order 32 = 25

Direct product of C2 and Q16

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

Aliases: C2xQ16, C4.8D4, C4.3C23, C8.5C22, C22.16D4, Q8.1C22, (C2xC8).4C2, C2.13(C2xD4), (C2xQ8).4C2, (C2xC4).28C22, SmallGroup(32,41)

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

Derived series
C1 — C4 — C2xQ16
Chief series
C1 — C2 — C4 — C2xC4 — C2xQ8 — C2xQ16
Lower central
C1 — C2 — C4 — C2xQ16
Upper central
C1 — C22 — C2xC4 — C2xQ16
Jennings
C1 — C2 — C2 — C4 — C2xQ16

Generators and relations for C2xQ16
 G = < a,b,c | a2=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 38 in 30 conjugacy classes, 22 normal (8 characteristic)
Quotients: C1, C2, C22, D4, C23, Q16, C2xD4, C2xQ16
C1
C2
C2
C2
C4
C4
C22
2C4
2C4
2C4
2C4
Q8
Q8
C8
Q8
Q8
C8
C2xC4
2C2xC4
2Q8
2Q8
2C2xC4
Q16
Q16
Q16
Q16
C2xC8
C2xQ8
C2xQ8
C2xQ16

Character table of C2xQ16

 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 C2xQ16
►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)]])
 

C2xQ16 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  Q8oD8  C3:S3:Q16
C2xQ16 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 C2xQ16 ►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] >;
 

C2xQ16 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 C2xQ16 in TeX
Character table of C2xQ16 in TeX

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