C₇×#13670 #13190886

order	1792
nilpotency class	5
isoclinism	isoclinic class of #13669
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13670

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#2184
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₇
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₇×#131 ◁ C₇×#504 ◁ G

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

normal subgroups only

This group has 278 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) = 71
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 71 points:

(1,2,3,4,5,6,7)(9,16)(10,21)(11,25)(12,14)(13,27)(17,39)(18,44)(19,45)(20,38)(22,49)(23,53)(24,31)(26,33)(28,56)(29,55)(30,57)(32,34)(35,37)(36,40)(41,50)(42,64)(43,63)(46,60)(47,51)(52,70)(54,65)(58,62)(59,69)(66,68)(67,71);(8,9)(10,17)(11,26)(12,18)(13,33)(14,35)(15,20)(16,25)(19,27)(21,40)(22,50)(23,54)(24,43)(28,45)(29,60)(30,58)(31,42)(32,46)(34,62)(36,51)(37,55)(38,57)(39,49)(41,48)(44,56)(47,65)(52,71)(53,66)(59,68)(61,67)(63,70)(64,69);(8,10,15,24)(9,17,20,43)(11,22,30,52)(12,31,14,21)(13,23,34,53)(16,36,38,64)(18,42,35,40)(19,41,46,67)(25,51,57,69)(26,50,58,71)(27,48,32,61)(28,59,29,47)(33,54,62,66)(37,63,44,39)(45,68,60,65)(49,55,70,56)

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, 6658);