D₅×GL(2,𝔽₅) #72277769

order	4800
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

non-solvable

Aliases

direct product

D₅×GL(2,𝔽₅)

Splittings

derived

C₅×SL(2,𝔽₅) ⋊ C₂×C₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₅×S₅
commutator, hypocenter	C₅×SL(2,𝔽₅)	C₂×C₄
Fitting	C₄×C₅	C₂×S₅
socle	C₂×C₅	C₂×A₅⋊C₄
prime radical	D₅×#6732	C₂
cosocle	C₅×#6732	C₂²
LCS	G ▷ C₅×SL(2,𝔽₅)
UCS	C₁ ◁ C₄

340 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 22 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) = 29
transitive degree	μ_t(G) = 120

A minimal faithful permutation representation — 2 generators on 29 points:

(1,5)(2,3)(6,15,25,13,23,28,20,14)(7,9,24,29,12,11,18,27)(8,10,21,17,26,19,16,22);(1,2)(3,4)(6,28,16,11,12)(7,23,15,21,9)(8,27,18,25,14)(13,26,29,24,20)

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×GL(2,𝔽₅)
maximal normal subgroup of	AGL(1,𝔽₅)×GL(2,𝔽₅)

GAP

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

gap> G := Group((1,5)(2,3)(6,15,25,13,23,28,20,14)(7,9,24,29,12,11,18,27)(8,10,21,17,26,19,16,22), (1,2)(3,4)(6,28,16,11,12)(7,23,15,21,9)(8,27,18,25,14)(13,26,29,24,20));