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G = Q8⋊C4  order 32 = 25

1st semidirect product of Q8 and C4 acting via C4/C2=C2

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

Aliases: Q8⋊1C4, C4.12D4, C2.1Q16, C2.2SD16, C22.9D4, C4⋊C4.1C2, C4.2(C2×C4), (C2×C8).1C2, (C2×Q8).2C2, C2.7(C22⋊C4), (C2×C4).15C22, SmallGroup(32,10)

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

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

Generators and relations for Q8⋊C4
 G = < a,b,c | a4=c4=1, b2=a2, bab-1=cac-1=a-1, cbc-1=a-1b >

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

Character table of Q8⋊C4

 class 12A2B2C4A4B4C4D4E4F8A8B8C8D
 size 11112244442222
ρ111111111111111    trivial
ρ2111111-11-11-1-1-1-1    linear of order 2
ρ3111111-1-1-1-11111    linear of order 2
ρ41111111-11-1-1-1-1-1    linear of order 2
ρ51-1-111-11-i-1i-i-iii    linear of order 4
ρ61-1-111-1-1-i1iii-i-i    linear of order 4
ρ71-1-111-1-1i1-i-i-iii    linear of order 4
ρ81-1-111-11i-1-iii-i-i    linear of order 4
ρ92222-2-200000000    orthogonal lifted from D4
ρ102-2-22-2200000000    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
ρ132-22-2000000√-2-√-2√-2-√-2    complex lifted from SD16
ρ142-22-2000000-√-2√-2-√-2√-2    complex lifted from SD16

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

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

Q8⋊C4 is a maximal subgroup of
 C23.24D4  C23.36D4  C23.38D4  C4×SD16  C4×Q16  SD16⋊C4  Q16⋊C4  Q8⋊D4  D4⋊D4  C22⋊Q16  D4.7D4  D4.D4  C4⋊2Q16  D4.2D4  Q8.D4  C8⋊8D4  C8⋊D4  Q8⋊Q8  C4.Q16  Q8.Q8  C23.47D4  C23.48D4  C23.20D4  C4.SD16  C42.78C22  C42.28C22  C42.30C22  Q8⋊Dic3  Q8⋊F5  C3⋊S3.2Q16  C62.4D4  C3⋊S3.5Q16  PSU3(𝔽2)⋊C4  D13.Q16
 C4p.D4: C8.18D4  C8.D4  C6.SD16  C2.Dic12  Q8⋊2Dic3  C10.Q16  C20.44D4  Q8⋊Dic5 ...
Q8⋊C4 is a maximal quotient of
 C23.31D4  C22.4Q16  Q8⋊F5  C3⋊S3.2Q16  C62.4D4  C3⋊S3.5Q16  PSU3(𝔽2)⋊C4  D13.Q16
 C2p.Q16: Q8⋊C8  C4.10D8  C4.6Q16  C6.SD16  C2.Dic12  Q8⋊2Dic3  C10.Q16  C20.44D4 ...

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

100
001
0160
,
100
017
0716
,
400
0114
046
G:=sub<GL(3,GF(17))| [1,0,0,0,0,16,0,1,0],[1,0,0,0,1,7,0,7,16],[4,0,0,0,11,4,0,4,6] >;
 

Q8⋊C4 in GAP, Magma, Sage, TeX 

Q_8\rtimes C_4
 
% in TeX
 
G:=Group("Q8:C4");
 
// GroupNames label
 
G:=SmallGroup(32,10);
 
// by ID
 
G=gap.SmallGroup(32,10);
 
# by ID
 
G:=PCGroup([5,-2,2,-2,2,-2,40,61,86,302,157,72]);
 
// Polycyclic
 
G:=Group<a,b,c|a^4=c^4=1,b^2=a^2,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^-1*b>;
 
// generators/relations
 

Export

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

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