C₇×#12447 #13189663

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12416
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#12447

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#459
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×C₈² ◁ G

234 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 896

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

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

Conjugacy classes & character table

301 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 5435);