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C₅×C₃⋊(C₃⋊C₈) #67106

order	360
isoclinism	isoclinic class of C₃⋊D₃
automorphism	?
homology	Abelianization (H₁) = C₅×C₈ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₅×C₃⋊(C₃⋊C₈)
unambiguous semidirect product	C₃⋊(C₅×C₃⋊C₈)

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₄×C₅	C₃⋊D₃
commutator, hypocenter	C₃²	C₅×C₈
Frattini	C₄	C₅×C₃⋊D₃
Fitting	C₃²×C₄×C₅	C₂
socle, prime radical	C₂×C₃²×C₅	C₄
cosocle	C₃²×C₄	C₂×C₅
LCS	G ▷ C₃²
UCS	C₁ ◁ C₄×C₅

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 — 3 generators on 19 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#13085586, #13085587, #13085588, …
maximal subgroup of	C₅×#3708, #10623724, D₅×C₃⋊(C₃⋊C₈), …
maximal normal subgroup of	C₅×#3708, #10623724, D₅×C₃⋊(C₃⋊C₈), …
maximal quotient of	C₅×C₃⋊(C₃⋊C₁₆), C₂×C₅×C₃⋊(C₃⋊C₈), C₅×C₃⋊(C₉⋊C₈), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 36);