C₇×#10670 #13187886

order	1792
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#10670

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
socle	C₂²×C₇	#399
prime radical	C₂³×C₇	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄²×C₇ ◁ G

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

normal subgroups only

This group has 316 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) = 448

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

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

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 3658);