C₇×#10613 #13187829

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10611
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#10613

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#396
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂³×C₇	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

276 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 98 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 — 3 generators on 39 points:

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

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 3601);