C₁₃×#7431 #15693770

order	3328
nilpotency class	5
isoclinism	isoclinic class of #7408
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₃×#7431

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃	AΣL(1,𝔽₁₆)
commutator	C₂×C₄⋊C₄	C₂×C₄×C₁₃
Frattini, cosocle	#391	C₂²×C₁₃
prime radical	C₂³×C₁₃	#99
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂³×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₁₃×#391 ◁ G

206 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 37
transitive degree	μ_t(G) = 416

A minimal faithful permutation representation — 2 generators on 37 points:

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

Conjugacy classes & character table

403 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 419);