AGL(1,𝔽₅)×D₁₁² #72228565

order	9680
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

AGL(1,𝔽₅)×D₁₁²

Splittings

Fitting, socle, derived, Hall	C₅×C₁₁² ⋊ C₂²×C₄
Sylow	C₁₁² ⋊ C₂²×AGL(1,𝔽₅)
Sylow	C₅ ⋊ C₄×D₁₁²

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅×C₁₁²	C₂²×C₄
prime radical	D₅×D₁₁²	C₂
cosocle	D₅×C₁₁²	C₂³
LCS	G ▷ C₅×C₁₁²
UCS	C₁

296 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 110

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

Conjugacy classes & character table

245 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)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26), (2,3,4,5)(6,7,9,11,13,15,16,14,12,10,8), (18,19)(20,21)(22,23)(24,25)(26,27));