C₇×#10517 #13187733

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10512
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#10517

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂³×C₇	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

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

normal subgroups only

This group has 344 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) = 55
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 55 points:

(1,2,3,4,5,6,7)(8,9)(10,18)(11,16)(12,15)(13,17)(14,20)(19,23)(21,22)(24,25,29,33)(26,34,37,49)(27,38,41,31)(28,42,44,32)(30,43,46,36)(35,51,39,50)(40,48,55,53)(45,52,47,54);(8,10)(9,14)(11,20)(12,21)(13,19)(15,23)(16,18)(17,22)(24,26,29,37)(25,30,33,46)(27,39,41,35)(28,43,44,36)(31,47,38,45)(32,49,42,34)(40,52,55,54)(48,50,53,51);(9,15)(10,19)(12,13)(16,17)(20,21)(22,23)(24,27,28,40,29,41,44,55)(25,31,32,48,33,38,42,53)(26,35,36,52,37,39,43,54)(30,45,34,50,46,47,49,51)

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 3505);