C₇×#13681 #13190897

order	1792
nilpotency class	5
isoclinism	isoclinic class of #13681
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13681

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#2196
commutator, Frattini, cosocle	C₄⋊Q₈	C₂³×C₇
LCS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂²×C₇ ◁ C₂×C₄×C₇ ◁ C₇×C₄⋊Q₈ ◁ G

398 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 58 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 — 3 generators on 23 points:

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

Conjugacy classes & character table

175 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6669);