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C₃×#1204 #67134

order	360
isoclinism	isoclinic class of #1201
automorphism	Aut(G) = C₂³×D₃×AGL(1,𝔽₅) (order 960) · Inn(G) = D₃×D₅ · Out(G) = C₂²×C₄
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₁

Aliases

Splittings

Hall	C₃²×C₅ ⋊ Q₈
Sylow	C₃² ⋊ Dic₁₀
Sylow	C₅ ⋊ C₃×Dic₆

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	D₃×D₅
commutator, cosocle	C₂×C₃×C₅	C₂²×C₃
Frattini	C₂	C₃×D₃×D₅
Fitting, socle, prime radical	C₂×C₃²×C₅	C₂²
hypocenter	C₃×C₅	C₃×Q₈
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₃

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 120

A minimal faithful permutation representation — 2 generators on 19 points:

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

Conjugacy classes & character table

63 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#1204, #10624038, #10624039, …
maximal normal subgroup of	D₃×#1204, #10624038, #10624039, …
maximal quotient of	C₃×#6668, C₃×#6669, C₃×Dic₃⋊Dic₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 64);