C₇×#13458 #13190674

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

Aliases

direct product

C₇×#13458

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

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

normal subgroups only

This group has 324 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) = 43
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 43 points:

(1,2,3,4,5,6,7)(12,13,15,19)(14,24)(16,30,29,40)(17,23,18,22)(20,36)(21,32,35,33)(25,39)(26,34)(27,41)(28,42)(31,37)(38,43);(8,9)(10,11)(13,19)(14,25,28,27)(20,37,34,38)(21,35)(22,23)(24,39,42,41)(26,43,36,31)(30,40);(9,11)(12,14,17,27,15,28,18,25)(13,20,22,37,19,34,23,38)(16,31,32,36,29,43,33,26)(21,39,40,42,35,41,30,24)

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, 6446);