C₃×C₂₃×D₂₅ #16812282

order	3450
isoclinism	isoclinic class of D₂₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₂₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×C₂₃×D₂₅
unambiguous semidirect product	C₂₅⋊(C₂×C₃×C₂₃)

Splittings

Fitting, Hall	C₃×C₂₃×C₂₅ ⋊ C₂
derived, Sylow	C₂₅ ⋊ C₂×C₃×C₂₃
Hall	C₂₃×C₂₅ ⋊ C₂×C₃
Hall	C₃×C₂₅ ⋊ C₂×C₂₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₂₃	D₂₅
commutator, cosocle, hypocenter	C₂₅	C₂×C₃×C₂₃
Frattini	C₅	C₃×D₅×C₂₃
Fitting	C₃×C₂₃×C₂₅	C₂
socle	C₃×C₅×C₂₃	D₅
LCS	G ▷ C₂₅
UCS	C₁ ◁ C₃×C₂₃

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 1725

A minimal faithful permutation representation — 2 generators on 51 points:

(1,2,3)(4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51);(28,51)(29,50)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃×C₇×C₂₃×D₂₅, C₃×C₂₃×D₁₇₅, ΓL(1,𝔽₈)×C₂₃×D₂₅, …
maximal normal subgroup of	C₃×C₇×C₂₃×D₂₅, C₃×C₁₁×C₂₃×D₂₅, C₃×D₂₅×C₂₃⋊C₁₁
maximal quotient of	C₃×C₇×C₂₃×D₂₅, C₃×C₂₃×D₁₇₅, ΓL(1,𝔽₈)×C₂₃×D₂₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3450, 3);

C3×C23×D25 #16812282