C₇×#13586 #13190802

order	1792
nilpotency class	5
isoclinism	isoclinic class of #13584
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂×C₄²

Aliases

direct product

C₇×#13586

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	C₂×#353
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	#127	C₂³×C₇
prime radical	C₇×#532	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#532 ◁ G

406 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 90 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)(10,19,21,14)(11,15)(12,17,27,16)(13,18)(20,35,36,28)(22,29,26,30)(23,37,38,31)(24,33,39,32)(25,34);(8,10,13,22)(9,14,18,30)(11,23,25,20)(12,26,27,21)(15,31,34,28)(16,29,17,19)(24,36,39,38)(32,35,33,37);(8,11,13,25)(9,15,18,34)(10,20,22,23)(12,24,27,39)(14,28,30,31)(16,32,17,33)(19,35,29,37)(21,36,26,38)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6574);