C₇×#32868 #13210084

order	1792
nilpotency class	3
isoclinism	isoclinic class of #32712
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#32868

Subgroups

	subgroup	G/H
center, socle	C₂³×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂²×C₇×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₇×#376 ◁ G

596 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 206 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) = 896

A minimal faithful permutation representation — 4 generators on 39 points:

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

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, 25856);