C₇×#13437 #13190653

order	1792
nilpotency class	4
isoclinism	isoclinic class of #13436
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13437

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#2044
commutator, Frattini, cosocle	C₂×#41	C₂³×C₇
prime radical	C₇×#535	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×#41 ◁ G

422 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 86 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)(8,9,14,19)(10,20,24,37)(11,15,13,17)(12,26,27,18)(16,31,33,23)(21,35,22,36)(25,34,28,30)(29,38,32,39);(9,15)(10,21)(11,13)(12,27)(17,19)(18,34)(22,24)(23,38)(26,30)(29,32)(31,39)(35,36);(8,10,11,22,14,24,13,21)(9,16,17,32,19,33,15,29)(12,23,25,39,27,31,28,38)(18,20,34,35,26,37,30,36)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6425);