C₁₃×#32716 #15719055

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

Aliases

direct product

C₁₃×#32716

Subgroups

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₁₃×#2615	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂²×C₁₃×#41 ◁ G

906 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 906 conjugacy classes of subgroups — too many to draw. Only its 206 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 — 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)(30,31,34,38)(32,39,41,36)(33,43,35,37)(40,45,42,44);(14,16)(15,18)(17,21)(19,20)(22,24)(23,26)(25,28)(27,29)(30,32,34,41)(31,36,38,39)(33,42,35,40)(37,44,43,45);(14,16)(15,19)(17,21)(18,20)(23,27)(26,29)(32,40)(33,35)(36,44)(37,43)(39,45)(41,42);(16,21)(18,19)(24,28)(26,29)(30,33,34,35)(31,37,38,43)(32,40,41,42)(36,45,39,44)

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, 25704);