C₇×#13676 #13190892

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

Aliases

direct product

C₇×#13676

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#2196
commutator, Frattini, cosocle	#127	C₂³×C₇
prime radical	C₇×#3291	C₂
LCS	G ▷ #127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂²×C₇ ◁ C₂×C₄×C₇ ◁ C₇×#127 ◁ G

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

normal subgroups only

This group has 394 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) = 39
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 39 points:

(1,2,3,4,5,6,7)(8,9,13,17)(10,14,20,28)(11,15,23,31)(12,24,25,16)(18,32,35,38)(19,27,36,37)(21,29,22,30)(26,34,39,33);(8,10,13,20)(9,14,17,28)(11,18,23,35)(12,19,25,36)(15,26,31,39)(16,27,24,37)(21,33,22,34)(29,32,30,38);(8,11)(9,15)(10,18)(12,21)(13,23)(14,26)(16,29)(17,31)(19,33)(20,35)(22,25)(24,30)(27,32)(28,39)(34,36)(37,38)

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, 6664);