C₇×#32873 #13210089

order	1792
nilpotency class	3
isoclinism	isoclinic class of #32873
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

direct product

C₇×#32873

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#2840
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₇
prime radical	C₂³×C₇×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂⁴×C₇ ◁ G

854 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 854 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 23
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 4 generators on 23 points:

(1,2,3,4,5,6,7)(8,9)(10,16)(11,19)(12,15)(13,18)(14,20)(17,22)(21,23);(8,10)(9,13,15,16)(11,17)(12,18)(14,22,19,21)(20,23);(8,11)(9,14,15,19)(10,17)(12,20)(13,21,16,22)(18,23);(17,23)(21,22)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 25861);