C₇×#12313 #13189529

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12310
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#12313

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1619
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₇×#3293	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₄×C₇×C₈ ◁ G

356 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 78 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)(8,9,14,10)(11,21,24,37)(12,26,31,27)(13,20,29,17)(15,32,18,23)(16,35,19,30)(22,36,25,38)(28,33,39,34);(8,10,14,9)(11,18,24,15)(12,27,31,26)(13,17,29,20)(16,30,19,35)(21,28,37,39)(22,33,25,34)(23,38,32,36);(8,11)(9,15)(10,18)(12,28)(13,22)(14,24)(16,36)(17,33)(19,38)(20,34)(21,26)(23,30)(25,29)(27,37)(31,39)(32,35)

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, 5301);