C₇×#7040 #13184256

order	1792
nilpotency class	3
isoclinism	isoclinic class of #7035
automorphism	?
homology	Abelianization (H₁) = C₄²×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#7040

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	#95
commutator	C₂²×C₄	C₄²×C₇
Frattini, cosocle	C₄³	C₂²×C₇
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₄³×C₇ ◁ G

292 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 1792

A minimal faithful permutation representation — 2 generators on 39 points:

(1,2,3,4,5,6,7)(9,13)(10,11)(14,15)(16,17,19,21,20,23,18,22)(24,25,27,30,28,31,35,38)(26,32,34,39,33,29,36,37);(8,9,11,14,12,15,10,13)(17,21)(18,19)(22,23)(24,26)(25,29,31,32)(27,34)(28,33)(30,37,38,39)(35,36)

Conjugacy classes & character table

364 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 28);