C₇×#12873 #13190089

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12873
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#12873

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1823
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₂×C₇×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×#131 ◁ G

380 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 224

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

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

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, 5861);