C₇×#9068 #13186284

order	1792
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#9068

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	#120
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
socle	C₂²×C₇	#386
prime radical	C₂³×C₇	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄²×C₇ ◁ G

404 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 896

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

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

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, 2056);