C₇×#10385 #13187601

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10382
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#10385

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#393
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₇
prime radical	C₂⁴×C₇	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₄×C₇ ◁ G

396 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 90 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)(8,9,11,14)(10,16)(12,15,20,22)(13,21)(17,18)(19,23)(24,25)(26,29)(27,34,36,30)(28,31)(32,37,35,39)(33,38);(8,9)(10,17,19,13)(11,14)(12,15)(16,18,23,21)(20,22)(24,26,28,33)(25,29,31,38)(27,35)(30,39)(32,36)(34,37);(8,10)(9,13)(11,19)(12,21)(14,17)(15,23)(16,22)(18,20)(24,27,28,36)(25,30,31,34)(26,32,33,35)(29,37,38,39)

Conjugacy classes & character table

217 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 3373);