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C₅×#3756 #10623896

order	720
isoclinism	isoclinic class of #3753
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₃

Aliases

Splittings

Hall	C₃²×C₅ ⋊ Q₁₆
Sylow	C₃² ⋊ C₅×Q₁₆

Subgroups

	subgroup	G/H
center	C₂×C₅	#638
commutator, cosocle	C₃²×C₄	C₂²×C₅
Frattini	C₄	C₂×C₅×C₃⋊D₃
Fitting	C₃²×C₅×Q₈	C₂
socle, prime radical	C₂×C₃²×C₅	D₄
hypercenter	C₅×Q₈	C₃⋊D₃
hypocenter	C₃²	C₅×Q₁₆
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅ ◁ C₅×Q₈

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 720

A minimal faithful permutation representation — 3 generators on 27 points:

(1,2,3,4,5)(6,8,10,16)(7,11,13,20)(9,18,19,15)(12,21,17,14)(25,26,27);(6,7,10,13)(8,14,16,21)(9,17,19,12)(11,18,20,15)(23,24)(26,27);(22,23,24)

Conjugacy classes & character table

135 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₅×#3756, #13086991, #13086994, …
maximal normal subgroup of	D₅×#3756, #13086991, #13086994, …
maximal quotient of	C₅×#63634, C₅×#63637, C₅×#63662, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 365);