C₇×#10383 #13187599

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10382
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#10383

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#393
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₇
prime radical	C₂⁴×C₇	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₄×C₇ ◁ G

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

normal subgroups only

This group has 404 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) = 39
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 39 points:

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

Conjugacy classes & character table

217 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 3371);