H(3,𝔽₅) #1249

order	125
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₅)
automorphism	Aut(G) = AGL(2,𝔽₅) (order 12000) · Inn(G) = C₅² · Out(G) = GL(2,𝔽₅)
homology	Abelianization (H₁) = C₅² · Schur multiplier (H₂) = C₅²

Aliases

extraspecial	5₊¹⁺²
unitriangular	UT(3,𝔽₅)
Heisenberg	H(3,𝔽₅)
unambiguous semidirect product	C₅²⋊C₅

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₅	C₅²
LCS	G ▷ C₅ ▷ C₁
UCS	C₁ ◁ C₅ ◁ G
LECS	G ▷ C₅ ▷ C₁

15 conjugacy classes of subgroups.

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 25

A minimal faithful permutation representation — 2 generators on 25 points:

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

Polycyclic presentation:

5^3|b^a=bc

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(125, 3);