C₇×#10095 #13187311

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10088
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#10095

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₇×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

480 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 102 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)(10,14)(11,13)(12,16)(15,19)(17,22)(18,23)(20,21)(26,32)(27,34)(29,37)(30,35)(33,36)(38,39);(9,13)(11,19)(14,22)(15,16)(17,20)(21,23)(24,25,28,31)(26,29,33,38)(27,35,34,30)(32,39,36,37);(8,10,12,18)(9,14,16,23)(11,20,19,17)(13,21,15,22)(24,26)(25,29)(27,36)(28,33)(30,39)(31,38)(32,34)(35,37)

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, 3083);