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C₃⋊D₃×C₂₅ #90643

order	450
isoclinism	isoclinic class of C₃⋊D₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₂₅ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₃⋊D₃×C₂₅
unambiguous semidirect product	C₃⋊(D₃×C₂₅)

Splittings

Fitting, Hall	C₃²×C₂₅ ⋊ C₂
derived, Sylow	C₃² ⋊ C₂×C₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂₅	C₃⋊D₃
commutator, hypocenter	C₃²	C₂×C₂₅
Frattini	C₅	C₅×C₃⋊D₃
Fitting	C₃²×C₂₅	C₂
socle, cosocle	C₃²×C₅	C₂×C₅
prime radical	C₅×C₃⋊D₃	C₅
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂₅

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 225

A minimal faithful permutation representation — 3 generators on 31 points:

(2,3)(5,6)(7,8,10,13,17,9,11,14,18,22,12,15,19,23,26,16,20,24,27,29,21,25,28,30,31);(4,5,6);(1,2,3)

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

derived subgroup of	D₃²⋊D₂₅, #157⋊D₂₅, C₂₅⋊PSU(3,𝔽₂)
maximal subgroup of	C₂₅×#157, #11744620, C₃⋊D₃×D₂₅, …
maximal normal subgroup of	C₂₅×#157, #11744620, C₃⋊D₃×D₂₅, …
maximal quotient of	C₂₅×C₃⋊Dic₃, C₂×C₃⋊D₃×C₂₅, C₂₅×C₃⋊D₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(450, 8);