C₇×#10392 #13187608

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10390
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇×#10392

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#393
commutator, Frattini, cosocle	C₂⁵	C₂³×C₇
prime radical	C₂³×C₇×D₄	C₂²
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂⁵×C₇ ◁ G

746 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 746 conjugacy classes of subgroups — too many to draw. Only its 90 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)(18,21,25,30)(19,26)(22,28)(24,31)(27,29);(8,10)(9,12)(11,14)(13,15)(16,18)(17,21)(19,27,28,24)(20,25)(22,29,26,31)(23,30);(12,15)(16,19)(17,22)(18,24)(20,28)(21,29)(23,26)(25,27)(30,31)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 3380);