C₇×#8744 #13185960

order	1792
nilpotency class	3
isoclinism	isoclinic class of #8738
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#8744

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#381
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂⁴×C₇	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

384 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 110 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)(10,17,19,13)(11,16)(12,15)(14,18,21,23)(20,22)(24,25,27,31,28,30,34,37)(26,29,32,36,33,35,38,39);(8,10)(9,13)(11,18)(12,19)(14,16)(15,17)(20,23)(21,22)(24,26,28,33)(25,29,30,35)(27,32,34,38)(31,36,37,39);(8,10)(9,14,15,21)(11,18)(12,19)(13,16,17,22)(20,23)(25,30)(27,34)(29,35)(32,38)

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, 1732);