C₃×#33546 #10652153

order	768
nilpotency class	4
isoclinism	isoclinic class of #2196
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

	subgroup	G/H
center	C₃×C₄	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×D₄	C₂⁴×C₃
socle	C₂×C₃	C₂×B(4,𝔽₂)
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₄ ◁ C₂×C₃×#41 ◁ G

1236 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 4 generators on 19 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 26534);