C₇×#13459 #13190675

order	1792
nilpotency class	4
isoclinism	isoclinic class of #13459
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#13459

Subgroups

	subgroup	G/H
center	C₄×C₇	#498
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂×C₇	#2058
prime radical	C₂³×C₇	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×#97 ◁ G

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

normal subgroups only

This group has 318 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)(10,19,21,36)(11,24)(12,17)(13,29)(14,18)(15,22,31,38)(16,28)(20,33,23,35)(25,39)(26,30,37,34)(27,32);(10,20,21,23)(11,25)(15,30,31,34)(16,32)(19,35,36,33)(22,26,38,37)(24,39)(27,28);(8,10,13,20,14,21,12,23)(9,15,17,34,18,31,29,30)(11,26,28,22,25,37,27,38)(16,33,24,19,32,35,39,36)

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, 6447);