C₇×#10036 #13187252

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#10036

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
socle	C₂²×C₇	#392
prime radical	C₂²×C₇×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄²×C₇ ◁ G

528 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 122 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)(8,9,13,10)(11,14,12,15)(16,17)(18,24)(19,27)(20,23)(21,26)(22,29)(25,31)(28,30);(9,14)(10,15)(11,12)(21,30)(24,31);(8,10,13,9)(11,15,12,14)(16,18)(17,21)(19,28)(20,25)(22,31)(23,30)(24,27)(26,29)

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, 3024);