C₇×#20792 #13198008

order	1792
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₇ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₇×#20792

Subgroups

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

1118 conjugacy classes of subgroups; only the 622 normal subgroups are shown.

normal subgroups only

This group has 1118 conjugacy classes of subgroups — too many to draw. Only its 622 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) = 896

A minimal faithful permutation representation — 4 generators on 31 points:

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

Conjugacy classes & character table

448 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 13780);