C₃×D₇₅ #90642

order	450
isoclinism	isoclinic class of D₇₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×D₇₅
unambiguous semidirect product	C₃⋊(C₃×D₂₅)
unambiguous semidirect product	C₂₅⋊(C₃×D₃)

Splittings

Fitting, Hall	C₃²×C₂₅ ⋊ C₂
derived	C₃×C₂₅ ⋊ C₂×C₃
Sylow	C₂₅ ⋊ C₃×D₃
Sylow	C₃² ⋊ D₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₃	D₇₅
commutator, cosocle, hypocenter	C₃×C₂₅	C₂×C₃
Frattini	C₅	C₃×D₁₅
Fitting	C₃²×C₂₅	C₂
socle	C₃²×C₅	D₅
LCS	G ▷ C₃×C₂₅
UCS	C₁ ◁ C₃

27 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 150

A minimal faithful permutation representation — 2 generators on 31 points:

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

Conjugacy classes & character table

117 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×D₃×D₂₅, D₃×D₇₅, (C₃⋊D₃)⋊D₂₅, …
maximal normal subgroup of	C₃×D₃×D₂₅, D₃×D₇₅, (C₃⋊D₃)⋊D₂₅, …
maximal quotient of	C₃×Dic₇₅, C₂×C₃×D₇₅, C₉×D₇₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(450, 7);

C3×D75 #90642

Aliases

Splittings

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₃×D₇₅ #90642