C₁₃×#11539 #15697878

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×#11539

Subgroups

	subgroup	G/H
center	C₄×C₁₃	#410
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×M₁₆	C₂³×C₁₃
socle	C₂×C₁₃	#1467
prime radical	C₂³×C₁₃	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃ ◁ C₂×C₈×C₁₃ ◁ C₁₃×#405 ◁ G

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

normal subgroups only

This group has 270 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)(14,15,18,24,19,23,34,40,20,25,32,42,17,21,31,36)(16,26,28,43,29,39,44,41,30,35,33,45,27,22,37,38);(15,21,25,23)(18,32)(22,35,39,26)(24,36,42,40)(28,33)(31,34)(37,44)(38,45,41,43);(14,16,19,29,20,30,17,27)(15,22,21,35,25,39,23,26)(18,33,34,37,32,28,31,44)(24,41,36,43,42,38,40,45)

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, 4527);