C₁₃×#12968 #15699307

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

Aliases

direct product

C₁₃×#12968

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

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

normal subgroups only

This group has 424 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)(14,15,19,24,20,25,18,32)(16,26,28,36,29,43,33,42)(17,23,30,22,31,39,34,40)(21,35,37,45,38,44,41,27);(14,16,19,28,20,29,18,33)(15,21,24,37,25,38,32,41)(17,27,30,35,31,45,34,44)(22,36,39,43,40,42,23,26);(14,17)(15,22)(16,27)(18,34)(19,30)(20,31)(21,36)(23,32)(24,39)(25,40)(26,41)(28,35)(29,45)(33,44)(37,43)(38,42)

Conjugacy classes & character table

520 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 5956);