C₁₃×#11538 #15697877

order	3328
nilpotency class	4
isoclinism	isoclinic class of #11522
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₃×#11538

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

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

normal subgroups only

This group has 334 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)(14,15,17,22,18,23,30,38,19,20,29,33,31,34,42,45)(16,24,26,40,27,32,43,37,25,39,28,44,41,21,35,36);(15,20)(16,25)(23,34)(26,28)(27,41)(35,43)(36,37)(40,44);(14,16,18,27,19,25,31,41)(15,21,34,39,20,32,23,24)(17,28,30,35,29,26,42,43)(22,37,45,40,33,36,38,44)

Conjugacy classes & character table

442 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 4526);