C₃⋊#674 #6645

order	240
isoclinism	isoclinic class of ΓL(1,𝔽₁₆)
automorphism	Aut(G) = C₄×D₃×AGL(1,𝔽₅) (order 480) · Inn(G) = ΓL(1,𝔽₁₆) · Out(G) = C₂×C₄
homology	Abelianization (H₁) = C₁₆ · Schur multiplier (H₂) = C₁

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₄	ΓL(1,𝔽₁₆)
commutator, hypocenter	C₃×C₅	C₁₆
Fitting	C₃×C₄×C₅	C₄
socle, prime radical	C₂×C₃×C₅	C₈
cosocle	C₃×#168	C₂
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₄

20 conjugacy classes of subgroups.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 240

A minimal faithful permutation representation — 2 generators on 24 points:

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

Polycyclic presentation:

2^4,3,5|a^2=b,b^2=c,c^2=d,e^a=e^2,f^a=f^3,f^b=f^4

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 6);