C₃×#37649 #10656256

order	768
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃ · Schur multiplier (H₂) = C₂⁹

Aliases

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

3222 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)(12,13,15,16)(14,18,19,17);(5,7)(13,16)(17,18);(4,5)(6,7)(8,9)(10,11)(12,13)(14,17)(15,16)(18,19);(12,14,15,19)(13,17,16,18);(4,5)(6,7)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 30637);