C₃×#455 #5329

order	192
nilpotency class	3
isoclinism	isoclinic class of #453
automorphism	Aut(G) = C₂²×#2196 (order 512) · Inn(G) = #120 · Out(G) = C₂⁴
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

	subgroup	G/H
center, socle	C₂×C₃	#120
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₃
prime radical	C₃×H(5,𝔽₂)	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ G

152 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 3 generators on 19 points:

(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(17,18,19);(1,3)(2,6)(4,13)(5,11)(7,16)(8,15)(9,12)(10,14);(4,14)(7,12)(9,16)(10,13)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 888);