C₃×#61155 #10679762

order	768
nilpotency class	3
isoclinism	isoclinic class of D₄×AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃ · Schur multiplier (H₂) = C₂⁸

Aliases

	subgroup	G/H
center, socle	C₂²×C₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄×D₄ ◁ G

2572 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 5 generators on 19 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 54143);