GO⁺(4,𝔽₂)⋊C₄ #64193

order	288
isoclinism	isoclinic class of AΓL(1,𝔽₉)
automorphism	Aut(G) = C₂²×AΓL(1,𝔽₉) (order 576) · Inn(G) = AΓL(1,𝔽₉) · Out(G) = C₂²
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

C₃² ⋊ #102

	subgroup	G/H
center, Frattini, hypercenter	C₂	AΓL(1,𝔽₉)
commutator	#157	C₂×C₄
Fitting, socle	C₂×C₃²	SD₁₆
prime radical	C₂×GO⁺(4,𝔽₂)	C₂
cosocle	C₂×#157	C₂²
hypocenter	C₃²	#102
LCS	G ▷ #157 ▷ C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂

66 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 2 generators on 13 points:

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

Polycyclic presentation:

2^5,3^2|a^2=c,c^2=e,b^a=bcde,f^a=g,g^a=fg^2,c^b=ce,f^b=fg^2,g^b=g^2,f^c=fg^2,g^c=f^2g^2,f^e=f^2,g^e=g^2

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 841);