C₇×#8741 #13185957

order	1792
nilpotency class	3
isoclinism	isoclinic class of #8738
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#8741

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#381
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂³×C₄	C₂³×C₇
prime radical	C₂⁴×C₇	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₄×C₇ ◁ G

418 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 448

A minimal faithful permutation representation — 3 generators on 39 points:

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

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 1729);