C₇×#8556 #13185772

order	1792
nilpotency class	3
isoclinism	isoclinic class of #8541
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇×#8556

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#379
commutator	C₂⁴	C₂²×C₄×C₇
Frattini, cosocle	C₂³×C₄	C₂³×C₇
prime radical	C₂⁵×C₇	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₄×C₇ ◁ G

842 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 842 conjugacy classes of subgroups — too many to draw. Only its 150 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)(8,9,12,19,13,17,11,24)(10,20,21,34,22,15,27,31)(14,18,28,33,25,32,26,16)(23,36,38,35,37,30,39,29);(8,10,14,23)(9,15,18,30)(11,21,26,38)(12,27,28,39)(13,22,25,37)(16,29,24,31)(17,20,32,36)(19,34,33,35);(8,11,25,28)(9,16,32,19)(10,21,37,39)(12,14,26,13)(15,29,36,34)(17,33,18,24)(20,35,30,31)(22,27,23,38)

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, 1544);