C₇×#23805 #13201021

order	1792
nilpotency class	3
isoclinism	isoclinic class of #23391
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇×#23805

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₄³×C₇ ◁ G

974 conjugacy classes of subgroups; only the 286 normal subgroups are shown.

normal subgroups only

This group has 974 conjugacy classes of subgroups — too many to draw. Only its 286 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) = 35
transitive degree	μ_t(G) = 1792

A minimal faithful permutation representation — 4 generators on 35 points:

(1,2,3,4,5,6,7)(12,13,15,16)(14,17,18,19)(20,21,24,29,26,28,25,30)(22,31,32,35,23,34,33,27);(8,9)(10,11)(13,16)(17,19)(20,22,26,23)(21,27,28,35)(24,33,25,32)(29,34,30,31);(9,11)(20,23,26,22)(21,27,28,35)(24,32,25,33)(29,34,30,31);(12,14,15,18)(13,17,16,19)(21,28)(27,35)(29,30)(31,34)

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, 16793);