C₇×#13340 #13190556

order	1792
nilpotency class	4
isoclinism	isoclinic class of #13340
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#13340

Subgroups

	subgroup	G/H
center	C₄×C₇	#493
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂×C₇	#2011
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×#97 ◁ G

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

normal subgroups only

This group has 552 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) = 224

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

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

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, 6328);