AGL(1,𝔽₅)×#11925892 #72228546

order	24200
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

AGL(1,𝔽₅)×#11925892

Splittings

Fitting, socle, derived	C₅×C₁₁² ⋊ C₂×C₄×C₅
Hall	C₅×#10597243 ⋊ C₂×C₄
Sylow	C₁₁² ⋊ C₂×C₅×AGL(1,𝔽₅)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅×C₁₁²	C₂×C₄×C₅
prime radical	D₅×#11925892	C₂
cosocle	D₅×C₁₁²	C₂²×C₅
LCS	G ▷ C₅×C₁₁²
UCS	C₁

476 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#11925892⋊C₂
maximal normal subgroup of	AGL(1,𝔽₅)×#11925892⋊C₂

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