C₄²×C₄⋊C₄ #13748

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) order 3145728
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂³×C₄³

Aliases

direct product	C₄²×C₄⋊C₄
unambiguous semidirect product	C₄⋊C₄³

	subgroup	G/H
center	C₂²×C₄²	C₂²
commutator	C₂	C₂×C₄³
Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁴
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1239 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 16 points:

(1,2,3,4);(5,6,7,8);(10,12)(13,14,15,16);(9,10,11,12)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6736);