C₅×C₃⋊#1113 #13108558

order	1620
isoclinism	isoclinic class of C₃⋊#1113
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product	C₅×C₃⋊#1113
unambiguous semidirect product	C₃⋊(C₅×#1113)

Fitting, socle, Hall	C₃⁴×C₅ ⋊ C₂²
derived, Sylow	C₃⁴ ⋊ C₂²×C₅

	subgroup	G/H
center, hypercenter	C₅	C₃⋊#1113
commutator, cosocle, hypocenter	C₃⁴	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₃⁴×C₅	C₂²
LCS	G ▷ C₃⁴
UCS	C₁ ◁ C₅

448 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 180

A minimal faithful permutation representation — 3 generators on 17 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1620, 464);