C₇×#13328 #13190544

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

Aliases

direct product

C₇×#13328

Subgroups

	subgroup	G/H
center, socle	C₂³×C₇	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₂×C₇×#493	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

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

normal subgroups only

This group has 662 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) = 31
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 3 generators on 31 points:

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

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, 6316);