Finite Group Databasebeta

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D₃×D₇×C₂₅ #14523545

order	2100
isoclinism	isoclinic class of D₃×D₇
automorphism	?
homology	Abelianization (H₁) = C₂²×C₂₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	D₃×D₇×C₂₅
unambiguous semidirect product	(D₃×C₇)⋊(C₂×C₂₅)
unambiguous semidirect product	(C₃×D₇)⋊(C₂×C₂₅)
unambiguous semidirect product	D₂₁⋊(C₂×C₂₅)

Splittings

Fitting, Hall	C₃×C₇×C₂₅ ⋊ C₂²
derived, Hall	C₃×C₇ ⋊ C₂²×C₂₅
Hall	C₇×C₂₅ ⋊ C₂×D₃
Hall	C₃×C₂₅ ⋊ C₂×D₇
Sylow	C₇ ⋊ C₂×D₃×C₂₅
Sylow	C₃ ⋊ C₂×D₇×C₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂₅	D₃×D₇
commutator, hypocenter	C₃×C₇	C₂²×C₂₅
Frattini	C₅	C₅×D₃×D₇
Fitting	C₃×C₇×C₂₅	C₂²
socle, cosocle	C₃×C₅×C₇	C₂²×C₅
prime radical	C₅×D₃×D₇	C₅
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₂₅

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 525

Conjugacy classes & character table

375 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃×C₂₅×D₄₉, D₃×C₇×D₇×C₂₅, D₃×C₂₅×C₇⋊D₇, …
maximal normal subgroup of	D₃×C₇×D₇×C₂₅, D₃×C₁₁×D₇×C₂₅, D₃×C₁₃×D₇×C₂₅, …
maximal quotient of	D₃×C₂₅×D₄₉, D₃×C₇×D₇×C₂₅, D₃×C₂₅×C₇⋊D₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2100, 25);