C₇×#15190 #13192406

order	1792
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁷

Aliases

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₇	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₇
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇ ◁ G

2278 conjugacy classes of subgroups; only the 590 normal subgroups are shown.

normal subgroups only

This group has 2278 conjugacy classes of subgroups — too many to draw. Only its 590 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

A minimal faithful permutation representation — 4 generators on 27 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 8178);