C₃²×D₁₇ #64560

order	306
isoclinism	isoclinic class of D₁₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃

Aliases

direct product	C₃²×D₁₇
unambiguous semidirect product	C₁₇⋊(C₂×C₃²)

Fitting, socle, Hall	C₃²×C₁₇ ⋊ C₂
derived, Sylow	C₁₇ ⋊ C₂×C₃²

	subgroup	G/H
center, hypercenter	C₃²	D₁₇
commutator, cosocle, hypocenter	C₁₇	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₃²×C₁₇	C₂
LCS	G ▷ C₁₇
UCS	C₁ ◁ C₃²

24 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 153

A minimal faithful permutation representation — 2 generators on 23 points:

(4,5,6)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23);(1,2,3)(7,8,10,12,14,16,18,20,22,23,21,19,17,15,13,11,9)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(306, 5);