C₇×#12436 #13189652

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12436
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#12436

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1667
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₄×C₇×C₈ ◁ G

234 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 74 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)(9,14)(10,19)(11,15)(12,24)(17,35)(18,29)(20,21)(22,36)(23,37)(25,30)(26,31)(27,28)(32,33)(38,39);(8,9,15,28,12,17,25,29,13,18,30,35,24,27,11,14)(10,16,19,31,21,32,36,39,23,34,37,38,22,33,20,26);(8,10)(9,16)(11,20)(12,21)(13,23)(14,26)(15,19)(17,32)(18,34)(22,24)(25,36)(27,33)(28,31)(29,39)(30,37)(35,38)

Conjugacy classes & character table

259 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5424);