C₇×#10216 #13187432

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

Aliases

direct product

C₇×#10216

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₇×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

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

normal subgroups only

This group has 468 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) = 47
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 47 points:

(1,2,3,4,5,6,7)(8,9)(10,12)(11,14)(13,15)(16,17,22,27)(18,28,31,42)(19,32,35,40)(20,26,21,25)(23,36,38,30)(24,33,41,34)(29,37,45,39)(43,46,47,44);(9,12)(10,15)(13,14)(16,18)(17,23)(19,29)(20,36)(21,30)(22,31)(24,39)(25,42)(26,28)(27,38)(32,43)(33,46)(34,44)(35,45)(37,41)(40,47);(8,9,11,14)(10,13,15,12)(16,19,22,35)(17,24,27,41)(18,29,31,45)(20,34,21,33)(23,37,38,39)(25,32,26,40)(28,43,42,47)(30,46,36,44)

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, 3204);