C₇×#13607 #13190823

order	1792
nilpotency class	5
isoclinism	isoclinic class of #13605
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13607

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#2127
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	#124	C₂³×C₇
prime radical	C₇×#535	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#535 ◁ G

332 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 224

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

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

Conjugacy classes & character table

175 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6595);