C₇×#13330 #13190546

order	1792
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#13330

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

558 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 146 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)(16,17,22,27)(18,28,19,31)(20,34,39,47)(21,40,37,26)(23,32,24,29)(25,43,46,38)(30,45,33,44)(35,42,36,41);(9,11)(12,13)(14,15)(16,18,22,19)(17,23,27,24)(20,35,39,36)(21,33,37,30)(25,44,46,45)(26,42,40,41)(28,47,31,34)(29,43,32,38);(13,15)(16,19,22,18)(17,24,27,23)(20,36,39,35)(21,30,37,33)(25,45,46,44)(26,41,40,42)(28,34,31,47)(29,38,32,43)

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, 6318);