C₇×#11911 #13189127

order	1792
nilpotency class	4
isoclinism	isoclinic class of #1355
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#11911

Subgroups

	subgroup	G/H
center	C₄×C₇	#329
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	#97	C₂³×C₇
socle	C₂×C₇	C₂×#329
prime radical	C₇×#532	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×#97 ◁ G

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

normal subgroups only

This group has 318 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) = 224

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

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

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, 4899);