#100×5₋¹⁺² #72275765

order	4000
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅² · Schur multiplier (H₂) = C₂

Aliases

direct product

#100×5₋¹⁺²

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	C₅²×C₂²⋊C₄
commutator	C₂²×C₅	C₂×C₄×C₅²
Frattini, cosocle	C₂×C₄×C₅	C₂²×C₅²
prime radical	C₂×C₅²×D₄	C₂×C₅
LCS	G ▷ C₂²×C₅ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×5₋¹⁺² ◁ G

230 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 33
transitive degree	μ_t(G) = 200

A minimal faithful permutation representation — 2 generators on 33 points:

(1,2,3,6,4,5,7,8)(10,12,15,20,28)(13,17,23,16,21)(18,25,22,29,24)(26,33,32,31,30);(2,5)(3,7)(9,10,13,18,26,14,15,17,24,31,27,28,23,29,33,11,12,16,22,30,19,20,21,25,32)

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	#100×D₂₅⋊C₅, #100×AGL(1,ℤ/25)
maximal subgroup of	#100×D₂₅⋊C₅, #453×5₋¹⁺²
maximal normal subgroup of	#100×D₂₅⋊C₅, #453×5₋¹⁺²

GAP

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

gap> G := Group((1,2,3,6,4,5,7,8)(10,12,15,20,28)(13,17,23,16,21)(18,25,22,29,24)(26,33,32,31,30), (2,5)(3,7)(9,10,13,18,26,14,15,17,24,31,27,28,23,29,33,11,12,16,22,30,19,20,21,25,32));

#100×5−1+2 #72275765