C₇×#13675 #13190891

order	1792
nilpotency class	5
isoclinism	isoclinic class of #13673
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13675

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#2196
commutator, Frattini, cosocle	#127	C₂³×C₇
prime radical	C₇×#3292	C₂
LCS	G ▷ #127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂²×C₇ ◁ C₂×C₄×C₇ ◁ C₇×#127 ◁ G

408 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 23
transitive degree	μ_t(G) = 112

A minimal faithful permutation representation — 3 generators on 23 points:

(1,2,3,4,5,6,7)(8,9,13,17)(10,18,21,22)(11,12,16,23)(14,20,15,19);(8,10)(9,14)(11,19)(12,18)(13,21)(15,17)(16,20)(22,23);(8,10)(9,15)(11,22)(12,20)(13,21)(14,17)(16,18)(19,23)

Conjugacy classes & character table

175 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6663);