C₇×#13361 #13190577

order	1792
nilpotency class	4
isoclinism	isoclinic class of #13360
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#13361

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#2021
commutator, Frattini, cosocle	C₂×#41	C₂³×C₇
prime radical	C₇×#3021	C₂
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×#41 ◁ G

686 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 224

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

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6349);