C₇×#8913 #13186129

order	1792
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#8913

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	#120
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ G

428 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.

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

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 27 points:

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

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 1901);