C₂×C₃×#121 #5852

order	192
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	Aut(G) order 8192
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

	subgroup	G/H
center, socle	C₂³×C₃	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

426 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 194 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 13 points:

(1,2)(3,4,5);(7,9)(11,13);(1,2)(10,11,12,13);(1,2)(6,7)(8,9)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1411);