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G = C4.Q8  order 32 = 25

1st non-split extension by C4 of Q8 acting via Q8/C4=C2

p-group, metacyclic, nilpotent (class 3), monomial

Aliases: C8⋊2C4, C4.1Q8, C2.3SD16, C22.10D4, C4⋊C4.2C2, (C2×C8).6C2, C4.6(C2×C4), C2.3(C4⋊C4), (C2×C4).17C22, SmallGroup(32,13)

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

Derived series
C1 — C4 — C4.Q8
Chief series
C1 — C2 — C22 — C2×C4 — C2×C8 — C4.Q8
Lower central
C1 — C2 — C4 — C4.Q8
Upper central
C1 — C22 — C2×C4 — C4.Q8
Jennings
C1 — C2 — C2 — C2×C4 — C4.Q8

Generators and relations for C4.Q8
 G = < a,b,c | a4=1, b4=a2, c2=a-1b2, ab=ba, cac-1=a-1, cbc-1=b3 >

C1
C2
C2
C2
C4
C22
C4
4C4
4C4
C8
C8
C2×C4
2C2×C4
2C2×C4
C4⋊C4
C2×C8
C4⋊C4
C4.Q8

Character table of C4.Q8

 class 12A2B2C4A4B4C4D4E4F8A8B8C8D
 size 11112244442222
ρ111111111111111    trivial
ρ2111111-111-1-1-1-1-1    linear of order 2
ρ31111111-1-11-1-1-1-1    linear of order 2
ρ4111111-1-1-1-11111    linear of order 2
ρ51-11-11-1i-ii-i-11-11    linear of order 4
ρ61-11-11-1-i-iii1-11-1    linear of order 4
ρ71-11-11-1ii-i-i1-11-1    linear of order 4
ρ81-11-11-1-ii-ii-11-11    linear of order 4
ρ92222-2-200000000    orthogonal lifted from D4
ρ102-22-2-2200000000    symplectic lifted from Q8, Schur index 2
ρ1122-2-2000000-√-2√-2√-2-√-2    complex lifted from SD16
ρ122-2-22000000-√-2-√-2√-2√-2    complex lifted from SD16
ρ1322-2-2000000√-2-√-2-√-2√-2    complex lifted from SD16
ρ142-2-22000000√-2√-2-√-2-√-2    complex lifted from SD16

Smallest permutation representation of C4.Q8
►Regular action on 32 points
Generators in S32
(1 21 5 17)(2 22 6 18)(3 23 7 19)(4 24 8 20)(9 28 13 32)(10 29 14 25)(11 30 15 26)(12 31 16 27)

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

C4.Q8 is a maximal subgroup of
 D8⋊2C4  C23.25D4  M4(2)⋊C4  C4×SD16  Q16⋊C4  D8⋊C4  C8⋊8D4  C8⋊2D4  C8.D4  Q8⋊Q8  D4⋊2Q8  D4.Q8  Q8.Q8  C23.46D4  C23.19D4  C23.47D4  C23.20D4  C8⋊3Q8  C8⋊Q8  C40⋊C4  C62.6D4  C4.PSU3(𝔽2)  C8⋊(C32⋊C4)  F9⋊C4  D26.8D4
 C4p.Q8: C8.Q8  C8.5Q8  C12.Q8  C8⋊Dic3  C20.Q8  C40⋊6C4  C4.Dic14  C8⋊Dic7 ...
C4.Q8 is a maximal quotient of
 C8⋊2C8  C22.4Q16  C40⋊C4  C62.6D4  C4.PSU3(𝔽2)  C8⋊(C32⋊C4)  F9⋊C4  D26.8D4
 C4p.Q8: C8.Q8  C12.Q8  C8⋊Dic3  C20.Q8  C40⋊6C4  C4.Dic14  C8⋊Dic7  C4.Dic22 ...

Matrix representation of C4.Q8 ►in GL3(𝔽17) generated by

1600
0016
010
,
100
0512
055
,
400
0125
055
G:=sub<GL(3,GF(17))| [16,0,0,0,0,1,0,16,0],[1,0,0,0,5,5,0,12,5],[4,0,0,0,12,5,0,5,5] >;
 

C4.Q8 in GAP, Magma, Sage, TeX 

C_4.Q_8
 
% in TeX
 
G:=Group("C4.Q8");
 
// GroupNames label
 
G:=SmallGroup(32,13);
 
// by ID
 
G=gap.SmallGroup(32,13);
 
# by ID
 
G:=PCGroup([5,-2,2,-2,2,-2,40,61,26,302,72]);
 
// Polycyclic
 
G:=Group<a,b,c|a^4=1,b^4=a^2,c^2=a^-1*b^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^3>;
 
// generators/relations
 

Export

Subgroup lattice of C4.Q8 in TeX
Character table of C4.Q8 in TeX

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