C₇×#10516 #13187732

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10512
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#10516

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂³×C₇	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

364 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 47
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 47 points:

(1,2,3,4,5,6,7)(8,9)(10,13)(11,14)(12,15)(16,17,21,25)(18,26,29,41)(19,30,33,23)(20,34,36,24)(22,35,38,28)(27,43,31,42)(32,40,47,45)(37,44,39,46);(8,9)(10,14)(11,13)(12,15)(16,18,21,29)(17,22,25,38)(19,31,33,27)(20,35,36,28)(23,39,30,37)(24,41,34,26)(32,44,47,46)(40,42,45,43);(9,13)(10,11)(14,15)(16,19,20,32,21,33,36,47)(17,23,24,40,25,30,34,45)(18,27,28,44,29,31,35,46)(22,37,26,42,38,39,41,43)

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 3504);