C₃×#525 #5855

order	192
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	Aut(G) = C₂×#72215858 (order 2048) · Inn(G) = C₂³ · Out(G) = C₂³×#120
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

	subgroup	G/H
center	C₂×C₃×C₄	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃
socle	C₂²×C₃	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ G

330 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.

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

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 4 generators on 15 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1414);