C₇×#19710 #13196926

order	1792
nilpotency class	3
isoclinism	isoclinic class of #19697
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₇×#19710

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

772 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 230 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) = 896

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

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

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, 12698);