C₇×#8757 #13185973

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×#8757

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#381
commutator	C₂⁴	C₂²×C₄×C₇
Frattini, cosocle	C₂³×C₄	C₂³×C₇
prime radical	C₂⁴×C₇	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂³×C₄×C₇ ◁ G

438 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 31
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 31 points:

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

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, 1745);