C₇×#8751 #13185967

order	1792
nilpotency class	3
isoclinism	isoclinic class of #8746
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#8751

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#381
commutator	C₂⁴	C₂²×C₄×C₇
Frattini, cosocle	C₂³×C₄	C₂³×C₇
prime radical	C₂³×C₇×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₄×C₇ ◁ G

754 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 224

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

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

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, 1739);