C₇×#13663 #13190879

order	1792
nilpotency class	5
isoclinism	isoclinic class of #13661
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13663

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₇
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂²×C₇×C₈ ◁ C₂²×C₇×C₁₆ ◁ G

298 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 896

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

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

Conjugacy classes & character table

343 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6651);