C₇×#13678 #13190894

order	1792
nilpotency class	5
isoclinism	isoclinic class of #13677
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13678

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#2196
commutator, Frattini, cosocle	#124	C₂³×C₇
LCS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂²×C₇ ◁ C₂×C₄×C₇ ◁ C₇×#124 ◁ G

400 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 58 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)(11,22)(15,31)(16,30)(17,34)(18,23)(19,36)(20,26)(25,27)(29,32)(33,37)(35,39);(8,9)(10,15)(11,23)(12,17)(13,18)(14,22)(16,28)(19,37)(20,32)(21,33)(24,39)(25,34)(26,35)(27,30)(29,38)(31,36);(8,10,13,21)(9,15,18,33)(11,24,22,38)(12,26,28,20)(14,29,23,39)(16,32,17,35)(19,27,36,25)(30,31,34,37)

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6666);