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C₂×C₃²×Dic₅ #67163

order	360
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₃²×C₄ · Schur multiplier (H₂) = C₂×C₃

Aliases

C₂×C₃²×Dic₅

Splittings

derived, Sylow	C₅ ⋊ C₂×C₃²×C₄
Hall	C₃²×C₅ ⋊ C₂×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂²×C₃²	D₅
commutator, hypocenter	C₅	C₂×C₃²×C₄
Frattini	C₂	C₂×C₃²×D₅
Fitting, socle, prime radical	C₂²×C₃²×C₅	C₂
cosocle	C₂×C₅	C₂²×C₃²
LCS	G ▷ C₅
UCS	C₁ ◁ C₂²×C₃²

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 360

A minimal faithful permutation representation — 2 generators on 17 points:

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

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×Dic₃×Dic₅, C₃×(C₂×D₃)⋊Dic₅, C₃×#6667, …
maximal normal subgroup of	C₃×Dic₃×Dic₅, C₃×(C₂×D₃)⋊Dic₅, C₃×#6667, …
maximal quotient of	C₂×C₃²×#168, C₃²×#681, C₃²×C₄×Dic₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 93);