C₇×#10586 #13187802

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10078
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#10586

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ G

350 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 1792

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

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

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, 3574);