C₇×#32874 #13210090

order	1792
nilpotency class	3
isoclinism	isoclinic class of #32873
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

direct product

C₇×#32874

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#2840
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₇
prime radical	C₂²×C₇×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂⁴×C₇ ◁ G

688 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 39
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 4 generators on 39 points:

(1,2,3,4,5,6,7)(8,9)(10,18)(11,22)(12,16)(13,17)(14,21)(15,24)(19,28)(20,31)(23,36)(25,30)(26,33)(27,35)(29,37)(32,39)(34,38);(8,10)(9,14,17,18)(11,19)(12,20)(13,21)(15,28,22,26)(16,27,30,31)(23,34)(24,33)(25,35)(29,38,36,39)(32,37);(8,11)(9,15,17,22)(10,19)(12,23)(13,24)(14,26,18,28)(16,29,30,36)(20,34)(21,33)(25,37)(27,39,31,38)(32,35);(8,12,13,25)(9,16,17,30)(10,20,21,35)(11,23,24,37)(14,27,18,31)(15,29,22,36)(19,32,33,34)(26,38,28,39)

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, 25862);