C₂×C₃×H(5,𝔽₂) #5975

order	192
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	Aut(G) order 73728
homology	Abelianization (H₁) = C₂⁵×C₃ · Schur multiplier (H₂) = C₂⁹

Aliases

	subgroup	G/H
center, socle	C₂²×C₃	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁵×C₃
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

898 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 5 generators on 13 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1534);