C₁₃×#12723 #15699062

order	3328
nilpotency class	4
isoclinism	isoclinic class of #12721
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₃×#12723

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₃	#1782
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×#41	C₂³×C₁₃
prime radical	C₁₃×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂×C₄×C₁₃ ◁ C₄×C₁₃×#41 ◁ G

694 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 138 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)(16,25,28,40)(17,22,30,38)(18,23)(19,24)(20,26)(21,29,37,44)(27,33,31,45)(32,36)(34,41)(35,42)(39,43);(14,16,19,28)(15,20,24,35)(17,26,30,42)(18,31,32,27)(21,33,37,45)(22,25,38,40)(23,39,36,34)(29,43,44,41);(14,17,19,30)(15,21)(16,26,28,42)(18,29,32,44)(20,33)(22,23)(24,37)(25,39)(27,41,31,43)(34,40)(35,45)(36,38)

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, 5711);