C₁₃×#13052 #15699391

order	3328
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₃×#13052

Subgroups

	subgroup	G/H
center	C₄×C₁₃	B(4,𝔽₂)
commutator	C₂×D₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×#41	C₂³×C₁₃
socle	C₂×C₁₃	#1889
prime radical	C₁₃×#3018	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃ ◁ C₂×C₄×C₁₃ ◁ C₂×C₁₃×#41 ◁ G

624 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 114 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)(14,15,19,24,17,22,26,35)(16,25,29,41,27,20,34,37)(18,23,32,40,30,21,33,39)(28,38,42,45,31,36,43,44);(14,16,26,34,17,27,19,29)(15,20,35,41,22,25,24,37)(18,31,32,43,30,28,33,42)(21,38,39,45,23,36,40,44);(15,21)(16,27)(18,30)(20,36)(22,23)(24,40)(25,38)(29,34)(32,33)(35,39)(37,45)(41,44)

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, 6040);