C₁₃×#11537 #15697876

order	3328
nilpotency class	4
isoclinism	isoclinic class of #11521
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₃×#11537

Subgroups

	subgroup	G/H
center	C₄×C₁₃	#410
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×M₁₆	C₂³×C₁₃
socle	C₂×C₁₃	#1466
prime radical	C₁₃×#525	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃ ◁ C₂×C₈×C₁₃ ◁ C₁₃×#405 ◁ G

348 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 102 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)(14,15,17,22,18,23,30,38,19,20,29,33,31,34,41,45)(16,24,25,40,26,32,42,37,27,39,28,44,43,21,35,36);(15,20)(21,32)(23,34)(24,39);(14,16,18,26,19,27,31,43)(15,21,34,39,20,32,23,24)(17,28,30,35,29,25,41,42)(22,37,45,40,33,36,38,44)

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, 4525);