C₇×#9424 #13186640

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×#9424

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#386
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂×C₇×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

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

normal subgroups only

This group has 436 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) = 448

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

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

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, 2412);