D₃²⋊C₈ #63726

order	288
isoclinism	isoclinic class of GO⁺(4,𝔽₂)
automorphism	Aut(G) = C₂³×GO⁺(4,𝔽₂) (order 576) · Inn(G) = GO⁺(4,𝔽₂) · Out(G) = C₂³
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂²

Aliases

D₃²⋊C₈

C₃² ⋊ C₂²⋊C₈

	subgroup	G/H
center, Frattini, hypercenter	C₄	GO⁺(4,𝔽₂)
commutator	C₃⋊D₃	C₂×C₈
Fitting	C₃²×C₄	D₄
socle	C₂×C₃²	C₂²⋊C₄
prime radical	C₂×D₃²	C₄
cosocle	C₄×C₃⋊D₃	C₂²
hypocenter	C₃²	C₂²⋊C₈
LCS	G ▷ C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₄

88 conjugacy classes of subgroups.

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 2 generators on 14 points:

(1,3)(2,5,6)(7,8,9,11,10,12,13,14);(1,2)(3,6)(4,5)

Polycyclic presentation:

2^5,3^2|b^2=d,d^2=e,b^a=bc,f^a=f^2,f^b=g,g^b=f,f^c=f^2,g^c=g^2

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 374);