C₂×C₃×#123 #5854

order	192
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	Aut(G) order 8192
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

	subgroup	G/H
center, socle	C₂³×C₃	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃
prime radical	C₂²×C₃×D₄	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

342 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 178 normal subgroups are shown.

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

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 17 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1413);