C₁₃×#12234 #15698573

order	3328
nilpotency class	4
isoclinism	isoclinic class of #12096
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₃×#12234

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₁₃
prime radical	C₂²×D₄×C₁₃	D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂³×C₁₃ ◁ C₂×C₁₃×C₄⋊C₄ ◁ G

448 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 832

A minimal faithful permutation representation — 3 generators on 37 points:

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

Conjugacy classes & character table

364 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 5222);