C₁₃×#32710 #15719049

order	3328
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₃ · Schur multiplier (H₂) = C₂×C₄²

Aliases

direct product

C₁₃×#32710

Subgroups

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂⁵×C₁₃	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂⁴×C₄×C₁₃ ◁ G

984 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 984 conjugacy classes of subgroups — too many to draw. Only its 226 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) = 832

A minimal faithful permutation representation — 4 generators on 45 points:

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

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, 25698);