AGL(1,𝔽₅)×C₁₁⋊D₁₁ #72228544

order	4840
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂×C₁₁

Aliases

direct product

AGL(1,𝔽₅)×C₁₁⋊D₁₁

Splittings

Fitting, socle, derived, Hall	C₅×C₁₁² ⋊ C₂×C₄
Sylow	C₁₁² ⋊ C₂×AGL(1,𝔽₅)
Sylow	C₅ ⋊ C₄×C₁₁⋊D₁₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅×C₁₁²	C₂×C₄
prime radical	D₅×C₁₁⋊D₁₁	C₂
cosocle	D₅×C₁₁²	C₂²
LCS	G ▷ C₅×C₁₁²
UCS	C₁

224 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 605

A minimal faithful permutation representation — 3 generators on 27 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#11925892, AGL(1,𝔽₅)×D₁₁²
maximal normal subgroup of	AGL(1,𝔽₅)×#11925892, AGL(1,𝔽₅)×D₁₁²

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

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

AGL(1,𝔽5)×C11⋊D11 #72228544