C₃⋊#672 #6642

order	240
isoclinism	isoclinic class of D₁₅
automorphism	Aut(G) = C₂×C₄×D₃×AGL(1,𝔽₅) (order 960) · Inn(G) = D₁₅ · Out(G) = C₂×C₄²
homology	Abelianization (H₁) = C₁₆ · Schur multiplier (H₂) = C₁

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₈	D₁₅
commutator, hypocenter	C₃×C₅	C₁₆
Fitting, cosocle	C₃×C₅×C₈	C₂
socle, prime radical	C₂×C₃×C₅	C₈
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₈

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 3);