C₁₃×#12969 #15699308

order	3328
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₃×#12969

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	#1867
prime radical	C₁₃×#532	C₂²
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃ ◁ C₂×C₄×C₁₃ ◁ C₂×C₁₃×#41 ◁ G

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

normal subgroups only

This group has 416 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)(14,15,19,24,20,25,18,33)(16,26,29,40,30,43,34,37)(17,23,31,22,32,41,35,42)(21,36,38,27,39,44,45,28);(14,16,19,29,20,30,18,34)(15,21,24,38,25,39,33,45)(17,28,31,36,32,27,35,44)(22,40,41,43,42,37,23,26);(14,17)(15,22)(16,27)(18,35)(19,31)(20,32)(21,37)(23,33)(24,41)(25,42)(26,38)(28,30)(29,44)(34,36)(39,40)(43,45)

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, 5957);