C₇×#13606 #13190822

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×#13606

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#2127
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	#124	C₂³×C₇
prime radical	C₇×#3221	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#536 ◁ G

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

normal subgroups only

This group has 320 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)(8,9,11,17)(10,15,20,29)(12,16,22,31)(13,18,23,32)(14,24,27,25)(19,30,33,39)(21,28,35,38)(26,36,37,34);(9,14)(11,22)(12,25)(15,28)(16,18)(20,35)(21,34)(23,24)(26,30)(27,31)(29,39)(33,36);(8,10,13,19)(9,15,18,30)(11,20,23,33)(12,21,25,34)(14,26,16,28)(17,29,32,39)(22,35,24,36)(27,37,31,38)

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, 6594);