C₂⁴×D₁₁ #67052

order	352
isoclinism	isoclinic class of D₁₁
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product	C₂⁴×D₁₁
unambiguous semidirect product	C₁₁⋊C₂⁵
unambiguous semidirect product	(C₂×C₁₁)⋊C₂⁴

Fitting, socle	C₂⁴×C₁₁ ⋊ C₂
derived, Sylow	C₁₁ ⋊ C₂⁵

	subgroup	G/H
center, hypercenter	C₂⁴	D₁₁
commutator, cosocle, hypocenter	C₁₁	C₂⁵
Frattini	C₁	G
Fitting, socle	C₂⁴×C₁₁	C₂
LCS	G ▷ C₁₁
UCS	C₁ ◁ C₂⁴

748 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 176

A minimal faithful permutation representation — 5 generators on 19 points:

(1,2)(3,4)(5,6)(7,8)(10,11)(12,13)(14,15)(16,17)(18,19);(7,8)(9,10,12,14,16,18,19,17,15,13,11);(5,6);(3,4);(1,2)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(352, 194);