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C₃²×#168 #67103

order	360
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₃²×C₈ · Schur multiplier (H₂) = C₃

Aliases

C₃²×#168

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₃²×C₄	D₅
commutator, hypocenter	C₅	C₃²×C₈
Frattini	C₄	C₃²×D₅
Fitting	C₃²×C₄×C₅	C₂
socle, prime radical	C₂×C₃²×C₅	C₄
cosocle	C₄×C₅	C₂×C₃²
LCS	G ▷ C₅
UCS	C₁ ◁ C₃²×C₄

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:

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

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃²×#674, C₃×C₃⋊#674, C₃⋊(C₃⋊#674), …
maximal normal subgroup of	C₃²×#674, C₃×C₃⋊#674, C₃⋊(C₃⋊#674), …
maximal quotient of	C₃²×#672, C₂×C₃²×#168, C₃×C₉×#168, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 33);