C₇×#10379 #13187595

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10378
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#10379

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#393
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂³×C₇	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

360 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 90 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)(9,14,16,28)(10,18)(11,23)(15,32,24,27)(17,31,29,19)(21,37)(22,35)(25,26)(30,33,36,39)(34,38);(9,15,16,24)(10,18)(14,27,28,32)(17,33,29,39)(19,36,31,30)(21,34)(22,35)(37,38);(8,9,13,17)(10,19,18,28)(11,24,26,39)(12,16,20,29)(14,22,31,35)(15,23,33,25)(21,27,34,30)(32,38,36,37)

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, 3367);