C₇×#10546 #13187762

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10370
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#10546

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	#124
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
socle	C₂²×C₇	#394
prime radical	C₇×#3150	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄²×C₇ ◁ G

472 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 114 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)(11,14)(13,15)(16,17)(18,25)(19,27)(20,28)(21,24)(22,26)(23,30)(29,31);(8,9)(10,12)(11,14)(13,15)(16,18)(17,22)(19,23)(20,29)(21,27)(24,31)(25,28)(26,30);(9,12)(10,15)(13,14)(16,19,20,22,21,25,30,31)(17,18,23,26,24,27,28,29)

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, 3534);