C₁₉×#396 #11926072

order	1216
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₉ · Schur multiplier (H₂) = C₂³

Aliases

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂³
commutator, Frattini, cosocle	C₂³	C₂³×C₁₉
prime radical	C₂⁴×C₁₉	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ G

186 conjugacy classes of subgroups.

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 608

A minimal faithful permutation representation — 3 generators on 31 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(24,25,26,27)(29,31);(21,23)(28,29,30,31);(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1216, 77);