C₇×#10213 #13187429

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10212
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇×#10213

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₇×#3253	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

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

normal subgroups only

This group has 498 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) = 448

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

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

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, 3201);