D₅×D₁₁² #72228548

order	4840
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

direct product

D₅×D₁₁²

Splittings

Fitting, socle, derived, Hall	C₅×C₁₁² ⋊ C₂³
Sylow	C₁₁² ⋊ C₂²×D₅
Sylow	C₅ ⋊ C₂×D₁₁²

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₁₁²	C₂³
LCS	G ▷ C₅×C₁₁²
UCS	C₁

178 conjugacy classes of subgroups.

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:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₅×#11925892⋊C₂, AGL(1,𝔽₅)×D₁₁²
maximal normal subgroup of	D₅×#11925892⋊C₂, AGL(1,𝔽₅)×D₁₁²

GAP

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

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

D5×D112 #72228548