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C₄×C₃⋊D₁₅ #67181

order	360
isoclinism	isoclinic class of C₃⋊D₁₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂×C₃

Aliases

C₄×C₃⋊D₁₅

Splittings

Fitting	C₃²×C₄×C₅ ⋊ C₂
derived, Hall	C₃²×C₅ ⋊ C₂×C₄
Sylow	C₃² ⋊ C₄×D₅
Sylow	C₅ ⋊ C₄×C₃⋊D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₄	C₃⋊D₁₅
commutator, hypocenter	C₃²×C₅	C₂×C₄
Frattini	C₂	C₂×C₃⋊D₁₅
Fitting	C₃²×C₄×C₅	C₂
socle, cosocle	C₂×C₃²×C₅	C₂²
prime radical	C₂×C₃⋊D₁₅	C₂
LCS	G ▷ C₃²×C₅
UCS	C₁ ◁ C₄

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 180

A minimal faithful permutation representation — 3 generators on 15 points:

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

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10623753, #10623756, #10623794, …
maximal normal subgroup of	#10623753, #10623756, #10623794, …
maximal quotient of	C₈×C₃⋊D₁₅, #10623899, C₄×C₃⋊Dic₁₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 111);