C₇×#10669 #13187885

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#10669

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
socle	C₂²×C₇	#399
prime radical	C₂³×C₇	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄²×C₇ ◁ G

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

normal subgroups only

This group has 296 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) = 55
transitive degree	μ_t(G) = 448

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

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

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, 3657);