C₇×#12851 #13190067

order	1792
nilpotency class	3
isoclinism	isoclinic class of #1828
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#12851

Subgroups

	subgroup	G/H
center	C₄×C₇	#381
commutator	C₂³	C₂²×C₇×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂×C₇	#1817
prime radical	C₂⁴×C₇	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂²×C₇×C₈ ◁ G

416 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 162 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)(8,9,11,16,12,17,23,30,13,14,22,25,24,26,35,36)(10,18,20,31,21,32,34,39,19,15,27,28,33,29,37,38);(9,14)(11,22)(15,18)(17,26)(20,27)(23,35)(29,32)(34,37);(8,10)(9,15,14,18)(11,20)(12,21)(13,19)(16,28,25,31)(17,29,26,32)(22,27)(23,34)(24,33)(30,38,36,39)(35,37)

Conjugacy classes & character table

406 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5839);