C₂²×C₇×3₋¹⁺² #10625504

order	756
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₇ · Schur multiplier (H₂) = C₂

Aliases

C₂²×C₇×3₋¹⁺²

	subgroup	G/H
center, socle	C₂²×C₃×C₇	C₃²
commutator, Frattini, cosocle	C₃	C₂²×C₃²×C₇
prime radical	C₂²×C₃²×C₇	C₃
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₇ ◁ G

80 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 252

A minimal faithful permutation representation — 2 generators on 20 points:

(1,2)(5,6,7,8,9,10,11)(12,13,15,14,17,19,18,16,20);(3,4)(13,16,17)(15,19,20)

308 conjugacy classes. The character table is too large to display.

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(756, 135);