C₇×#10557 #13187773

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10557
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#10557

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#394
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₇
prime radical	C₇×#3186	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₄×C₇ ◁ G

500 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 31
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 31 points:

(1,2,3,4,5,6,7)(8,9)(10,12)(11,13)(14,15)(16,17)(18,25)(19,21)(20,24)(22,26)(23,28)(27,30)(29,31);(8,10)(9,12)(11,14)(13,15)(17,21)(18,26)(19,28)(22,30)(23,24)(25,29);(12,15)(16,18,19,27,20,26,28,31)(17,22,23,30,24,25,21,29)

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 3545);