C₇×#13450 #13190666

order	1792
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13450

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

410 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 47
transitive degree	μ_t(G) = 1792

A minimal faithful permutation representation — 3 generators on 47 points:

(1,2,3,4,5,6,7)(8,9)(10,11)(16,17,24,31)(18,32,37,47)(19,38,35,45)(20,28,39,29)(21,27,22,42)(23,43,33,30)(25,41,46,34)(26,40,44,36);(12,13)(14,15)(16,18,24,37)(17,25,31,46)(19,39,35,20)(21,34,22,41)(23,40,33,36)(26,27,44,42)(28,32,29,47)(30,38,43,45);(9,11)(13,15)(16,19,20,18,33,34,22,40,24,35,39,37,23,41,21,36)(17,26,27,46,43,47,29,45,31,44,42,25,30,32,28,38)

Conjugacy classes & character table

364 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6438);