C₇×#8766 #13185982

order	1792
nilpotency class	3
isoclinism	isoclinic class of #8760
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#8766

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#381
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂³×C₇	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

422 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 142 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)(16,17,22,26,20,27,21,25)(18,23,28,24,19,29,30,31);(16,18,20,19)(17,23,27,29)(21,28,22,30)(24,26,31,25);(9,12)(10,15)(13,14)(16,19,20,18)(17,24)(21,30,22,28)(23,26)(25,29)(27,31)

Conjugacy classes & character table

301 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 1754);