C₁₃×#10369 #15696708

order	3328
nilpotency class	3
isoclinism	isoclinic class of #10362
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₃×#10369

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃	#393
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
prime radical	C₂³×C₁₃	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

364 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 832

A minimal faithful permutation representation — 3 generators on 45 points:

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

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, 3357);