Finite Group Databasebeta

← Prev (#13063408) Next (#13063410) →

C₂₃×ΓL(1,𝔽₁₆) #13063409

order	1380
isoclinism	isoclinic class of ΓL(1,𝔽₁₆)
automorphism	?
homology	Abelianization (H₁) = C₄×C₂₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₂₃×ΓL(1,𝔽₁₆)
unambiguous semidirect product	C₃⋊(AGL(1,𝔽₅)×C₂₃)

Splittings

Fitting, socle, Hall	C₃×C₅×C₂₃ ⋊ C₄
derived, Hall	C₃×C₅ ⋊ C₄×C₂₃
Hall	C₅×C₂₃ ⋊ Dic₃
Hall	C₃×C₂₃ ⋊ AGL(1,𝔽₅)
Sylow	C₅ ⋊ Dic₃×C₂₃
Sylow	C₃ ⋊ AGL(1,𝔽₅)×C₂₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂₃	ΓL(1,𝔽₁₆)
commutator, hypocenter	C₃×C₅	C₄×C₂₃
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₂₃	C₄
prime radical	C₃×D₅×C₂₃	C₂
cosocle	C₃×D₅	C₂×C₂₃
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₂₃

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 345

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

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

Conjugacy classes & character table

207 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂₃×C₉⋊AGL(1,𝔽₅), C₃×C₂₃×ΓL(1,𝔽₁₆), C₂₃×C₃⋊ΓL(1,𝔽₁₆), …
maximal normal subgroup of	C₃×C₂₃×ΓL(1,𝔽₁₆), ΓL(1,𝔽₁₆)×C₂₃⋊C₁₁, C₁₁×C₂₃×ΓL(1,𝔽₁₆), …
maximal quotient of	C₂₃×C₉⋊AGL(1,𝔽₅), C₃×C₂₃×ΓL(1,𝔽₁₆), C₂₃×C₃⋊ΓL(1,𝔽₁₆), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1380, 12);