C₇×#12061 #13189277

order	1792
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇×C₈ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#12061

Subgroups

	subgroup	G/H
center	C₂×C₇×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₇×C₈
Frattini, cosocle	C₄×C₈	C₂³×C₇
socle, prime radical	C₂²×C₇	C₈×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₈ ◁ C₄×C₇×C₁₆ ◁ G

200 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 114 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 — 3 generators on 39 points:

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

Conjugacy classes & character table

448 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5049);