C₇×#8759 #13185975

order	1792
nilpotency class	3
isoclinism	isoclinic class of #8746
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#8759

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#381
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

474 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 110 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)(8,9,12,19,14,18,10,20)(11,24,22,38,26,33,27,30)(13,17,28,31,21,29,23,15)(16,32,34,25,35,39,36,37);(8,10,21,28)(9,15,29,19)(11,22,37,32)(12,13,23,14)(16,30,24,36)(17,20,18,31)(25,39,26,27)(33,34,35,38);(8,11)(9,16,17,33)(10,22)(12,27)(13,25)(14,26)(15,30,20,34)(18,35,29,24)(19,36,31,38)(21,37)(23,39)(28,32)

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 1747);