C₇×#12392 #13189608

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12316
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#12392

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₇×#3334	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ C₇×C₈×D₄ ◁ G

324 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 896

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

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

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, 5380);