C₇×#10529 #13187745

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10524
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#10529

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂³×C₇	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

326 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 55
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 3 generators on 55 points:

(1,2,3,4,5,6,7)(8,9)(10,18)(11,16)(12,15)(13,17)(14,20)(19,23)(21,22)(24,25,30,34)(26,35,39,50)(27,40,43,32)(28,45,29,33)(31,38,47,37)(36,51,53,48)(41,46,52,55)(42,49,44,54);(8,10)(9,14)(11,20)(12,21)(13,19)(15,23)(16,18)(17,22)(24,26,30,39)(25,31,34,47)(27,41,43,52)(28,38,29,37)(32,48,40,51)(33,50,45,35)(36,42,53,44)(46,54,55,49);(9,15)(10,19)(12,13)(16,17)(20,21)(22,23)(24,27,28,42,30,43,29,44)(25,32,33,49,34,40,45,54)(26,36,37,52,39,53,38,41)(31,46,35,48,47,55,50,51)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 3517);