C₇×#10038 #13187254

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10038
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#10038

Subgroups

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

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

normal subgroups only

This group has 532 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)(9,14)(11,12)(15,22)(16,17)(18,23)(19,21)(24,25)(26,32)(27,35)(28,31)(29,34)(30,37)(33,39)(36,38);(8,9)(10,18)(11,17)(12,14)(13,16)(15,19)(20,22)(21,23)(29,38)(32,39);(8,10)(9,15)(11,19)(12,21)(13,20)(14,18)(16,23)(17,22)(24,26)(25,29)(27,36)(28,33)(30,39)(31,38)(32,35)(34,37)

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, 3026);