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C₃×#87832 #11925318

order	1200
isoclinism	isoclinic class of #87831
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂

Aliases

Splittings

Hall	C₃×C₅² ⋊ C₄⋊C₄
Sylow	C₅² ⋊ C₃×C₄⋊C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	D₅⋊AGL(1,𝔽₅)
commutator	C₂×C₅²	C₂×C₃×C₄
Frattini	C₂	C₃×D₅⋊AGL(1,𝔽₅)
Fitting, socle	C₂×C₃×C₅²	C₂×C₄
prime radical	C₂×C₃×C₅⋊D₅	C₂²
cosocle	C₂×C₅⋊D₅	C₂²×C₃
hypocenter	C₅²	C₃×C₄⋊C₄
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₃

136 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 60

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

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

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#11717375, D₃×#87832, #72224577, …
maximal normal subgroup of	C₃×#11717375, D₃×#87832, #72224577, …
maximal quotient of	C₃×#11925356, C₃²×#87832, C₂×C₃×#87832, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1200, 528);