C₇×#23566 #13200782

order	1792
nilpotency class	3
isoclinism	isoclinic class of #23444
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₇×#23566

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₇
prime radical	C₂⁴×C₇	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₄×C₇ ◁ G

1118 conjugacy classes of subgroups; only the 366 normal subgroups are shown.

normal subgroups only

This group has 1118 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) = 448

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

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

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, 16554);