C₂²×C₇×#500 #13237870

order	1792
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₇ · Schur multiplier (H₂) = C₂⁹

Aliases

C₂²×C₇×#500

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇ ◁ C₂²×C₄²×C₇ ◁ G

2398 conjugacy classes of subgroups.

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 1792

A minimal faithful permutation representation — 5 generators on 27 points:

(1,2)(21,22,23,24,25,26,27);(3,4);(5,6,8,10)(7,11,12,9)(14,17)(15,18)(19,20);(5,7,8,12)(6,9,10,11);(13,14,18,20,16,19,15,17)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 53642);