C₇×#13444 #13190660

order	1792
nilpotency class	4
isoclinism	isoclinic class of #13309
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13444

Subgroups

	subgroup	G/H
center	C₄×C₇	#469
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂×C₇	#2054
prime radical	C₇×#3414	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

392 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 106 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)(10,11)(12,25)(13,33)(14,35)(15,20)(16,21)(17,27)(18,32)(19,31)(22,37)(23,39)(24,29)(26,30)(28,36)(34,38);(8,10)(9,16)(11,25)(12,30)(13,17)(14,36)(15,24)(18,28)(19,26)(20,37)(21,32)(22,34)(23,31)(27,35)(29,33)(38,39);(8,11,12,26,31,39,34,37,15,29,13,27,14,28,32,16)(9,17,18,24,35,22,33,23,20,30,38,10,19,21,25,36)

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6432);