GO⁺(4,𝔽₂) #643

order	72
isoclinism	isoclinic class of GO⁺(4,𝔽₂)
automorphism	Aut(G) = AΓL(1,𝔽₉) (order 144) · Inn(G) = G · Out(G) = C₂
holomorph	G ⋊ Aut(G) = #72220466
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

general orthogonal	GO⁺(4,𝔽₂)
special orthogonal	SO⁺(4,𝔽₂)
unambiguous semidirect product	#157⋊C₂
unambiguous semidirect product	D₃²⋊C₂

C₃² ⋊ D₄

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle	C₃⋊D₃	C₂²
Fitting, socle, hypocenter	C₃²	D₄
LCS	G ▷ C₃⋊D₃ ▷ C₃²
UCS	C₁

26 conjugacy classes of subgroups.

minimal degree	μ(G) = 6
transitive degree	μ_t(G) = 6

A minimal faithful permutation representation — 2 generators on 6 points:

(1,3)(2,5,6);(1,2)(3,6)(4,5)

Polycyclic presentation:

2^3,3^2|b^a=bc,d^a=d^2,d^b=e,e^b=d,d^c=d^2,e^c=e^2

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(72, 40);