C₇×#10664 #13187880

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

Aliases

direct product

C₇×#10664

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
socle	C₂²×C₇	#399
prime radical	C₂³×C₇	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₄²×C₇ ◁ G

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

normal subgroups only

This group has 304 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) = 39
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 3 generators on 39 points:

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

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, 3652);