C₁₃×#13560 #15699899

order	3328
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₁₃×#13560

Subgroups

	subgroup	G/H
center	C₄×C₁₃	AΣL(1,𝔽₁₆)
commutator	C₂×D₄	C₂²×C₄×C₁₃
Frattini, cosocle	#127	C₂³×C₁₃
socle	C₂×C₁₃	C₂×AΣL(1,𝔽₁₆)
prime radical	C₁₃×#3018	C₂
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃ ◁ C₂×C₄×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₁₃×#532 ◁ G

486 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 90 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 — 3 generators on 45 points:

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

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, 6548);