C₇×#10215 #13187431

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

Aliases

direct product

C₇×#10215

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

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

normal subgroups only

This group has 464 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)(11,13)(12,22)(14,16)(15,18)(19,23)(20,21)(26,33)(27,28)(30,34)(31,37)(35,36)(38,39);(8,10,13,19)(9,15,11,21)(12,20,14,18)(16,23,22,17)(24,25)(26,34)(27,31)(28,37)(29,32)(30,36)(33,38)(35,39);(8,11,13,9)(10,17,19,23)(12,22,14,16)(15,18,21,20)(24,26)(25,30)(27,36)(28,33)(29,35)(31,39)(32,38)(34,37)

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, 3203);