C₇×#23897 #13201113

order	1792
nilpotency class	3
isoclinism	isoclinic class of #23897
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₇×#23897

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#2403
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂²×C₄×C₇ ◁ G

1952 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 1952 conjugacy classes of subgroups — too many to draw. Only its 282 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)(14,21)(17,23)(20,22);(13,22)(16,23);(8,10)(9,13)(11,20)(12,17)(14,23)(15,22)(16,19)(18,21);(11,21)(14,19)(16,23)(18,20)

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, 16885);