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C₂×#4379 #67219

order	360
isoclinism	isoclinic class of #4379
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂×C₃

Aliases

Splittings

Fitting, socle	C₂×C₃²×C₅ ⋊ C₄
derived, Hall	C₃²×C₅ ⋊ C₂×C₄
Sylow	C₃² ⋊ C₂×Dic₅
Sylow	C₅ ⋊ C₂×#157

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#4379
commutator, hypocenter	C₃²×C₅	C₂×C₄
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₅	C₄
prime radical	C₂×C₅×C₃⋊D₃	C₂
cosocle	C₅×C₃⋊D₃	C₂²
LCS	G ▷ C₃²×C₅
UCS	C₁ ◁ C₂

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 60

A minimal faithful permutation representation — 2 generators on 13 points:

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

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Dic₅×#157, #10623985, #10623986, …
maximal normal subgroup of	Dic₅×#157, #10623985, #10623986, …
maximal quotient of	#10624106, #10624107, C₄×#4379, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 149);