C₇×#10361 #13187577

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10078
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#10361

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ G

392 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 1792

A minimal faithful permutation representation — 3 generators on 39 points:

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

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, 3349);