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D₃×#11717375 #72224587

order	4800
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

D₃×#11717375

Splittings

Fitting	C₂²×C₃×C₅² ⋊ C₂²×C₄
Hall	C₃×C₅² ⋊ C₂×C₄×D₄
Sylow	C₅² ⋊ C₄×D₃×D₄
Sylow	C₃ ⋊ C₂×#11717375

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃×D₅⋊AGL(1,𝔽₅)
commutator, socle	C₂×C₃×C₅²	C₂³×C₄
Fitting	C₂²×C₃×C₅²	C₂²×C₄
prime radical	D₃×#87886	C₂
cosocle	C₂×C₃×C₅⋊D₅	C₂⁴
hypercenter	C₂²	D₃×D₅⋊AGL(1,𝔽₅)
hypocenter	C₃×C₅²	C₂×C₄×D₄
LCS	G ▷ C₂×C₃×C₅² ▷ C₃×C₅²
UCS	C₁ ◁ C₂ ◁ C₂²

2176 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 17 points:

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

Conjugacy classes & character table

114 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224650, D₃×#13107559, D₉×#11717375, …
maximal normal subgroup of	D₃×#13107559, C₃×D₃×#11717375, C₂×D₃×#11717375, …
maximal quotient of	#72224650, D₃×#13104059, D₃×#13104055, …
holomorph of	C₃×Dic₅, C₂×C₃×D₅

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2,3)(9,10,11,12)(13,14,15,16), (2,3)(8,9,12,10,11), (5,7), (4,5,6,7)(14,15)(16,17));