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G = C4⋊C4  order 16 = 24

The semidirect product of C4 and C4 acting via C4/C2=C2

p-group, metacyclic, nilpotent (class 2), monomial

Aliases: C4⋊C4, C2.Q8, C2.2D4, C22.3C22, C2.2(C2×C4), (C2×C4).1C2, SmallGroup(16,4)

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

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

Generators and relations for C4⋊C4
 G = < a,b | a4=b4=1, bab-1=a-1 >

C1
C2
C2
C2
C22
C4
C4
2C4
2C4
C2×C4
C2×C4
C2×C4
C4⋊C4

Character table of C4⋊C4

 class 12A2B2C4A4B4C4D4E4F
 size 1111222222
ρ11111111111    trivial
ρ21111-11-1-11-1    linear of order 2
ρ31111-1-11-1-11    linear of order 2
ρ411111-1-11-1-1    linear of order 2
ρ511-1-1-1-ii1i-i    linear of order 4
ρ611-1-11-i-i-1ii    linear of order 4
ρ711-1-1-1i-i1-ii    linear of order 4
ρ811-1-11ii-1-i-i    linear of order 4
ρ92-22-2000000    orthogonal lifted from D4
ρ102-2-22000000    symplectic lifted from Q8, Schur index 2

Permutation representations of C4⋊C4
►Regular action on 16 points - transitive group 16T8
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

(1 14 12 5)(2 13 9 8)(3 16 10 7)(4 15 11 6)
 
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,14,12,5)(2,13,9,8)(3,16,10,7)(4,15,11,6)>;
 
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,14,12,5)(2,13,9,8)(3,16,10,7)(4,15,11,6) );
 
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,14,12,5),(2,13,9,8),(3,16,10,7),(4,15,11,6)]])
 
G:=TransitiveGroup(16,8);
 

C4⋊C4 is a maximal subgroup of
 C42⋊C2  C4×D4  C4×Q8  C4⋊D4  C22.D4  C42.C2  C42⋊2C2  C3⋊S3.Q8  C2.PSU3(𝔽2)  C4⋊(C32⋊C4)  C2.D5≀C2  (C5×C10).Q8
 C4p⋊C4: C4.Q8  C2.D8  C4⋊Dic3  C4⋊Dic5  C4⋊F5  C4⋊Dic7  C44⋊C4  C52⋊3C4 ...
 C2p.D4: D4⋊C4  Q8⋊C4  C22⋊Q8  C4⋊Q8  Dic3⋊C4  C10.D4  Dic7⋊C4  Dic11⋊C4 ...
C4⋊C4 is a maximal quotient of
 C2.C42  C3⋊S3.Q8  C2.PSU3(𝔽2)  C4⋊(C32⋊C4)  C2.D5≀C2  (C5×C10).Q8
 C4p⋊C4: C4.Q8  C2.D8  C4⋊Dic3  C4⋊Dic5  C4⋊F5  C4⋊Dic7  C44⋊C4  C52⋊3C4 ...
 C2p.D4: C4⋊C8  C8.C4  Dic3⋊C4  C10.D4  Dic7⋊C4  Dic11⋊C4  C26.D4  C34.D4 ...

Matrix representation of C4⋊C4 ►in GL3(𝔽5) generated by

100
004
010
,
300
030
002
G:=sub<GL(3,GF(5))| [1,0,0,0,0,1,0,4,0],[3,0,0,0,3,0,0,0,2] >;
 

C4⋊C4 in GAP, Magma, Sage, TeX 

C_4\rtimes C_4
 
% in TeX
 
G:=Group("C4:C4");
 
// GroupNames label
 
G:=SmallGroup(16,4);
 
// by ID
 
G=gap.SmallGroup(16,4);
 
# by ID
 
G:=PCGroup([4,-2,2,-2,2,32,49,21]);
 
// Polycyclic
 
G:=Group<a,b|a^4=b^4=1,b*a*b^-1=a^-1>;
 
// generators/relations
 

Export

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

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