C₇×#8733 #13185949

order	1792
nilpotency class	3
isoclinism	isoclinic class of #8603
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇×#8733

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	#117
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
socle	C₂²×C₇	#381
prime radical	C₂⁴×C₇	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₇×#376 ◁ G

404 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 126 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)(24,25,28,31,27,30,35,38)(26,32,34,39,33,29,36,37);(8,10)(9,13)(11,18)(12,19)(14,16)(15,17)(20,23)(21,22)(24,26)(25,29)(27,33)(28,34)(30,32)(31,37)(35,36)(38,39);(8,10)(9,14,15,21)(11,18)(12,19)(13,16,17,22)(20,23)(26,33)(29,32)(34,36)(37,39)

Conjugacy classes & character table

364 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 1721);