C₇×#13457 #13190673

order	1792
nilpotency class	4
isoclinism	isoclinic class of #13371
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#13457

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#476
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₂³×C₇	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

356 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 27
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 27 points:

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

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, 6445);