C₇×#13439 #13190655

order	1792
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13439

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₂²×C₇×D₁₆	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#405 ◁ G

418 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 110 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)(8,9)(10,11)(12,13)(14,25)(15,19)(16,31)(17,34)(18,24)(20,26)(21,29)(22,32)(23,30)(27,42)(28,38)(33,37)(35,40)(36,43)(39,41);(13,19)(14,26)(15,29)(16,32)(17,30)(21,23)(22,34)(24,31)(25,35)(27,43)(28,33)(36,39)(38,42)(40,41);(9,11)(12,14,15,26,30,40,32,43,18,28,16,33,17,27,29,41)(13,20,21,25,34,39,31,42,24,37,22,38,23,36,19,35)

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, 6427);