C₇×#18727 #13195943

order	1792
nilpotency class	3
isoclinism	isoclinic class of #18712
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇×#18727

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

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

normal subgroups only

This group has 996 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) = 448

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

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

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, 11715);