C₇×#13855 #13191071

order	1792
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₇ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₇×#13855

Subgroups

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

1246 conjugacy classes of subgroups; only the 494 normal subgroups are shown.

normal subgroups only

This group has 1246 conjugacy classes of subgroups — too many to draw. Only its 494 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) = 35
transitive degree	μ_t(G) = 1792

A minimal faithful permutation representation — 4 generators on 35 points:

(1,2,3,4,5,6,7)(8,9,10,11)(12,13)(14,19)(15,17)(16,18)(20,21)(22,24)(23,25)(26,27);(12,14,16,15)(13,17,18,19)(20,22,23,26)(21,24,25,27)(28,29)(30,33)(31,32)(34,35);(12,15,16,14)(13,17,18,19)(21,25)(24,27)(29,32)(33,35);(28,30,31,34)(29,33,32,35)

Conjugacy classes & character table

616 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6843);