C₇×#13667 #13190883

order	1792
nilpotency class	5
isoclinism	isoclinic class of #13667
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13667

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 ◁ C₂²×C₇×C₈ ◁ C₂×C₇×M₃₂ ◁ G

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

normal subgroups only

This group has 294 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)(10,18)(11,21)(12,24)(15,27)(16,30)(17,31)(19,23)(20,32)(22,37)(25,36)(26,28)(29,38)(33,34)(35,39);(8,9)(10,19)(11,23)(12,26)(13,15)(14,18)(16,21)(17,24)(20,35)(22,38)(25,39)(27,32)(28,37)(29,36)(30,33)(31,34);(10,20)(14,27)(18,32)(19,35)(22,33)(28,31)(30,38)(34,37)

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, 6655);