C₁₃×#8865 #15695204

order	3328
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₃×#8865

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₃	#120
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
socle	C₂²×C₁₃	#385
prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

348 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 1664

A minimal faithful permutation representation — 3 generators on 45 points:

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

Conjugacy classes & character table

598 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 1853);