C₇×#19753 #13196969

order	1792
nilpotency class	3
isoclinism	isoclinic class of #19745
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₇×#19753

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₂×C₇×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

736 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 230 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 — 4 generators on 39 points:

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

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 12741);