C₁₃×#10015 #15696354

order	3328
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₃×#10015

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₃	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
socle	C₂²×C₁₃	#392
prime radical	C₁₃×#2397	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

696 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 416

A minimal faithful permutation representation — 3 generators on 37 points:

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

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, 3003);