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C₃²×#170 #67122

order	360
isoclinism	isoclinic class of AGL(1,𝔽₅)
automorphism	?
homology	Abelianization (H₁) = C₃²×C₈ · Schur multiplier (H₂) = C₃

Aliases

C₃²×#170

Splittings

derived, Sylow	C₅ ⋊ C₃²×C₈
Hall	C₃²×C₅ ⋊ C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃²	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₃²×C₈
Frattini	C₂	C₃²×AGL(1,𝔽₅)
Fitting, socle, prime radical	C₂×C₃²×C₅	C₄
cosocle	Dic₅	C₂×C₃²
LCS	G ▷ C₅
UCS	C₁ ◁ C₂×C₃²

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 360

A minimal faithful permutation representation — 2 generators on 19 points:

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×D₃×#170, C₃×#6738, C₃×#6739, …
maximal normal subgroup of	C₃×D₃×#170, C₃×#6738, C₃×#6739, …
maximal quotient of	C₃²×#674, C₂×C₃²×#170, C₃×C₉×#170, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 52);