C₇×#8777 #13185993

order	1792
nilpotency class	3
isoclinism	isoclinic class of #8760
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇×#8777

Subgroups

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

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

normal subgroups only

This group has 480 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) = 224

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

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

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, 1765);