C₇×#13682 #13190898

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

Aliases

direct product

C₇×#13682

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

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

normal subgroups only

This group has 384 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) = 39
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 39 points:

(1,2,3,4,5,6,7)(9,15)(10,19)(11,22)(12,25)(17,35)(18,23)(20,39)(21,27)(24,29)(26,37)(30,32)(31,38);(8,9,14,18)(10,16,21,34)(11,23,22,15)(12,26,25,35)(13,17,28,37)(19,38,27,31)(20,33,29,36)(24,32,39,30);(8,10)(9,16)(11,24)(12,27)(13,29)(14,21)(15,30)(17,36)(18,34)(19,25)(20,28)(22,39)(23,32)(26,31)(33,37)(35,38)

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, 6670);