C₃×C₅²×D₂₃ #16812295

order	3450
isoclinism	isoclinic class of D₂₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₅² · Schur multiplier (H₂) = C₅

Aliases

direct product	C₃×C₅²×D₂₃
unambiguous semidirect product	C₂₃⋊(C₂×C₃×C₅²)

Splittings

Fitting, socle, Hall	C₃×C₅²×C₂₃ ⋊ C₂
derived, Sylow	C₂₃ ⋊ C₂×C₃×C₅²
Hall	C₅²×C₂₃ ⋊ C₂×C₃
Hall	C₃×C₂₃ ⋊ C₂×C₅²

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₅²	D₂₃
commutator, cosocle, hypocenter	C₂₃	C₂×C₃×C₅²
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₂₃	C₂
LCS	G ▷ C₂₃
UCS	C₁ ◁ C₃×C₅²

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 1725

A minimal faithful permutation representation — 2 generators on 36 points:

(1,2,3)(4,5,6,7,8)(14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36);(9,10,11,12,13)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₅²×ΓL(1,𝔽₈)⋊D₂₃, C₅²×ΓL(1,𝔽₈)×D₂₃, C₃×C₅²×C₇×D₂₃, …
maximal normal subgroup of	C₃×C₅²×C₇×D₂₃, C₃×C₅²×AGL(1,𝔽₂₃), C₃×C₅²×C₁₁×D₂₃
maximal quotient of	C₅²×ΓL(1,𝔽₈)⋊D₂₃, C₅²×ΓL(1,𝔽₈)×D₂₃, C₃×C₅²×C₇×D₂₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3450, 16);

C3×C52×D23 #16812295