C₇×#9420 #13186636

order	1792
nilpotency class	3
isoclinism	isoclinic class of #9415
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#9420

Subgroups

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

384 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 118 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)(8,9,13,17,14,16,12,10)(11,18,19,15,20,23,21,22)(24,25,30,33,29,34,28,32)(26,35,37,31,38,39,36,27);(8,10,13,16,14,17,12,9)(11,18,19,22,20,23,21,15)(24,26)(25,31)(27,34)(28,37)(29,38)(30,36)(32,39)(33,35);(8,11,14,20)(9,15,16,22)(10,18,17,23)(12,21,13,19)(24,27,29,31)(25,26,34,38)(28,35,30,39)(32,37,33,36)

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 2408);