C₇×#13373 #13190589

order	1792
nilpotency class	4
isoclinism	isoclinic class of #13373
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13373

Subgroups

	subgroup	G/H
center	C₄×C₇	#486
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂×C₇	#2023
prime radical	C₂×C₇×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×#97 ◁ G

364 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 110 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)(10,19,21,36)(11,24)(12,17)(13,29)(14,18)(15,22,31,38)(16,28)(20,37,23,33)(25,35)(26,34,39,30)(27,32);(10,20,21,23)(11,25)(15,30,31,34)(16,32)(19,33,36,37)(22,39,38,26)(24,35)(27,28);(8,10)(9,15)(11,26)(12,23)(13,20)(14,21)(16,33)(17,34)(18,31)(19,35)(22,27)(24,36)(25,39)(28,38)(29,30)(32,37)

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, 6361);