C₇×#12339 #13189555

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12324
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#12339

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₂×C₇×D₈	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ C₇×#446 ◁ G

308 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 47
transitive degree	μ_t(G) = 896

A minimal faithful permutation representation — 3 generators on 47 points:

(1,2,3,4,5,6,7)(9,12)(10,14)(13,15)(16,17,21,25)(18,26,28,44)(19,30,32,46)(20,35,33,24)(22,37,38,27)(23,40,42,34)(29,45,36,41)(31,43,47,39);(10,14)(12,15)(16,18)(17,22)(19,31)(20,36)(21,28)(23,41)(24,43)(25,38)(26,30)(27,34)(29,33)(32,47)(35,39)(37,40)(42,45)(44,46);(8,9,11,13)(10,15,14,12)(19,32)(23,42)(26,44)(27,37)(30,46)(31,47)(34,40)(41,45)

Conjugacy classes & character table

217 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5327);