C₇×#10632 #13187848

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10631
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#10632

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#397
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

310 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 106 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)(16,17,21,26,22,27,20,25)(18,28,31,23,32,39,30,40)(19,33,35,41,36,42,37,24)(29,44,45,47,34,38,46,43);(8,9)(10,12)(11,14)(13,15)(16,18,22,32)(17,23,27,40)(19,34,36,29)(20,31,21,30)(24,38,41,44)(25,39,26,28)(33,47,42,43)(35,45,37,46);(9,12)(10,15)(13,14)(16,19,20,35,22,36,21,37)(17,24,25,33,27,41,26,42)(18,29,30,45,32,34,31,46)(23,38,28,43,40,44,39,47)

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, 3620);