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AGL(1,𝔽₅)×C₂₃ #90747

order	460
isoclinism	isoclinic class of AGL(1,𝔽₅)
automorphism	Aut(G) = C₂×C₁₁×AGL(1,𝔽₅) (order 440) · Inn(G) = AGL(1,𝔽₅) · Out(G) = C₂×C₁₁
homology	Abelianization (H₁) = C₄×C₂₃ · Schur multiplier (H₂) = C₁

Aliases

AGL(1,𝔽₅)×C₂₃

Splittings

Fitting, socle, Hall	C₅×C₂₃ ⋊ C₄
derived, Sylow	C₅ ⋊ C₄×C₂₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂₃	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₄×C₂₃
Frattini	C₁	G
Fitting, socle	C₅×C₂₃	C₄
prime radical	D₅×C₂₃	C₂
cosocle	D₅	C₂×C₂₃
LCS	G ▷ C₅
UCS	C₁ ◁ C₂₃

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 115

A minimal faithful permutation representation — 2 generators on 28 points:

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

Conjugacy classes & character table

115 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×D₂₃, C₂×AGL(1,𝔽₅)×C₂₃, C₃×AGL(1,𝔽₅)×C₂₃, …
maximal normal subgroup of	AGL(1,𝔽₅)×D₂₃, C₂×AGL(1,𝔽₅)×C₂₃, C₃×AGL(1,𝔽₅)×C₂₃, …
maximal quotient of	C₂₃×#170, C₂×AGL(1,𝔽₅)×C₂₃, C₃×AGL(1,𝔽₅)×C₂₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(460, 5);