C₁₃×#7374 #15693713

order	3328
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₁₃ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₁₃×#7374

Subgroups

	subgroup	G/H
center	C₄×C₁₃	#353
commutator	C₄²	C₂×C₈×C₁₃
Frattini, cosocle	#423	C₂²×C₁₃
socle	C₂×C₁₃	#1325
prime radical	C₂³×C₁₃	#101
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃ ◁ C₂×C₄×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₁₃×#423 ◁ G

148 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 46 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 — 2 generators on 45 points:

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

Conjugacy classes & character table

364 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 362);