C₇×#19743 #13196959

order	1792
nilpotency class	3
isoclinism	isoclinic class of #19733
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₇×#19743

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

814 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 230 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) = 448

A minimal faithful permutation representation — 4 generators on 39 points:

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

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, 12731);