C₁₃×#7164 #15693503

order	3328
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₁₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₃×#7164

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₃	#95
commutator	C₂×C₄	C₄×C₈×C₁₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₃
socle	C₂²×C₁₃	#336
prime radical	C₂³×C₁₃	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ 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) = 37
transitive degree	μ_t(G) = 1664

A minimal faithful permutation representation — 2 generators on 37 points:

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

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, 152);