C₇×#15947 #13193163

order	1792
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₇×#15947

Subgroups

	subgroup	G/H
center, socle	C₂⁴×C₇	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₇
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇ ◁ G

6146 conjugacy classes of subgroups; only the 494 normal subgroups are shown.

normal subgroups only

This group has 6146 conjugacy classes of subgroups — too many to draw. Only its 494 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) = 23
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 4 generators on 23 points:

(1,2,3,4,5,6,7)(8,9,10,11)(12,13,14,15)(16,17,18,19)(20,21,22,23);(13,15)(21,23);(16,17)(18,19)(20,21)(22,23);(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)

Conjugacy classes & character table

406 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₇×#15947, ΓL(1,𝔽₈)×#15947, AGL(1,𝔽₇)×#15947
maximal subgroup of	D₇×#15947, ΓL(1,𝔽₈)×#15947
maximal normal subgroup of	D₇×#15947, ΓL(1,𝔽₈)×#15947

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 8935);