C₇×#21795 #13199011

order	1792
nilpotency class	3
isoclinism	isoclinic class of #17405
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₇ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

direct product

C₇×#21795

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₇
Frattini, cosocle	C₂⁴	C₂⁴×C₇
prime radical	C₂×C₇×#534	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₇×C₂²⋊C₄ ◁ G

1538 conjugacy classes of subgroups; only the 366 normal subgroups are shown.

normal subgroups only

This group has 1538 conjugacy classes of subgroups — too many to draw. Only its 366 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) = 224

A minimal faithful permutation representation — 4 generators on 31 points:

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

Conjugacy classes & character table

322 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 14783);