C₃×#37650 #10656257

order	768
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

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

3946 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)(9,11)(13,14)(17,18);(5,7)(12,13,16,14)(15,18,19,17);(4,5)(6,7)(8,9)(10,11);(12,14,16,13)(15,18,19,17);(4,5)(6,7)(12,15)(13,17)(14,18)(16,19)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 30638);