C₇×#10324 #13187540

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10320
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#10324

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#392
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₇
prime radical	C₂⁵×C₇	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₄×C₇ ◁ G

910 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 224

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

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

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, 3312);