C₂₉×5₋¹⁺² #16813752

order	3625
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₅)
automorphism	?
homology	Abelianization (H₁) = C₅²×C₂₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₂₉×5₋¹⁺²
unambiguous semidirect product	C₂₅⋊(C₅×C₂₉)
unambiguous semidirect product	(C₂₅×C₂₉)⋊C₅

Subgroups

	subgroup	G/H
center, socle	C₅×C₂₉	C₅²
commutator, Frattini, cosocle	C₅	C₅²×C₂₉
prime radical	C₅²×C₂₉	C₅
LCS	G ▷ C₅ ▷ C₁
UCS	C₁ ◁ C₅×C₂₉ ◁ G

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 54
transitive degree	μ_t(G) = 725

A minimal faithful permutation representation — 2 generators on 54 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29)(30,31,33,37,44,32,35,39,46,51,36,42,48,45,52,43,41,38,47,53,50,34,40,49,54);(31,34,41,42,35)(33,38,39,40,48)(37,45,49,46,47)(44,51,52,53,54)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂₉×#10597831, C₂₉×#10597832, C₂₉×#10597833, …
maximal normal subgroup of	C₂₉×#10597831, C₂₉×#10597832, C₂₉×#10597833, …
maximal quotient of	C₂₉×C₅²⋊C₂₅, C₂₉×C₂₅⋊C₂₅, C₅×C₂₉×5₋¹⁺²

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3625, 4);

C29×5−1+2 #16813752