C₁₃×#18997 #15705336

order	3328
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₃ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₁₃×#18997

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃
prime radical	C₂×C₁₃×H(3,𝔽₄)	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

1066 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1066 conjugacy classes of subgroups — too many to draw. Only its 222 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) = 45
transitive degree	μ_t(G) = 416

A minimal faithful permutation representation — 4 generators on 45 points:

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

Conjugacy classes & character table

442 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 11985);