C₃×#60561 #10679168

order	768
nilpotency class	3
isoclinism	isoclinic class of #3003
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃ · Schur multiplier (H₂) = C₂⁸

Aliases

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃
socle	C₂²×C₃	C₂³×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄² ◁ G

2428 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 192

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

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

maximal subgroup of	C₃×#10576942
maximal normal subgroup of	C₃×#10576942

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 53549);