C₇×#10208 #13187424

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10208
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#10208

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

508 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 94 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 — 3 generators on 39 points:

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

Conjugacy classes & character table

175 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 3196);