C₇×#9419 #13186635

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×#9419

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#386
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₇×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

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

normal subgroups only

This group has 424 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)(8,9,12,17,13,15,11,16)(10,18,21,14,20,23,19,22)(24,25,29,33,30,32,28,26)(27,34,35,31,36,39,37,38);(8,10)(9,14)(11,21)(12,19)(13,20)(15,22)(16,23)(17,18)(24,26,29,32,30,33,28,25)(27,34,35,38,36,39,37,31);(9,15)(11,12)(14,22)(19,21)(24,27)(25,31)(26,34)(28,37)(29,35)(30,36)(32,38)(33,39)

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, 2407);