C₇×#9973 #13187189

order	1792
nilpotency class	3
isoclinism	isoclinic class of #9914
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#9973

Subgroups

	subgroup	G/H
center, socle	C₂³×C₇	#120
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₇
prime radical	C₂⁴×C₇	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂³×C₄×C₇ ◁ G

632 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 448

A minimal faithful permutation representation — 3 generators on 35 points:

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

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 2961);