C₇×#10380 #13187596

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

Aliases

direct product

C₇×#10380

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#393
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₇×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

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

normal subgroups only

This group has 390 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) = 31
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 31 points:

(1,2,3,4,5,6,7)(8,9)(10,12)(11,14)(13,15)(16,17,19,22)(18,25)(20,24,21,29)(23,26)(27,30)(28,31);(10,15)(12,13)(17,22)(18,26,28,27)(23,31,30,25)(24,29);(9,12)(10,15)(13,14)(16,18)(17,23)(19,28)(20,26)(21,27)(22,30)(24,31)(25,29)

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, 3368);