C₁₃×#12047 #15698386

order	3328
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₃×#12047

Subgroups

	subgroup	G/H
center	C₈×C₁₃	C₂×D₈
commutator	C₈	C₂²×C₈×C₁₃
Frattini, cosocle	C₄×C₈	C₂³×C₁₃
socle	C₂×C₁₃	C₈×D₈
prime radical	C₁₃×#508	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₃ ◁ C₂×C₁₃×C₁₆ ◁ C₄×C₁₃×C₁₆ ◁ G

212 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 106 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)(14,15,19,25,21,26,38,24,22,27,18,37,20,16,28,30)(17,31,33,45,35,29,43,44,36,23,32,40,34,41,42,39);(14,16)(15,21)(17,29)(18,24)(19,30)(20,27)(22,26)(23,35)(25,38)(28,37)(31,34)(32,39)(33,44)(36,41)(40,43)(42,45);(14,17,19,33,21,35,38,43,22,36,18,32,20,34,28,42)(15,23,25,40,26,41,24,39,27,31,37,45,16,29,30,44)

Conjugacy classes & character table

832 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 5035);