C₁₃×#12724 #15699063

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×#12724

Subgroups

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

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

normal subgroups only

This group has 670 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)(17,29,31,22)(18,23)(19,24)(20,26,36,43)(21,30,37,45)(27,40)(28,41)(32,34)(33,38)(35,42,39,44);(14,16,19,28)(15,20,24,36)(17,26,31,43)(18,32,33,27)(21,34,37,40)(22,25,29,41)(23,39,38,35)(30,44,45,42);(14,17,19,31)(15,21)(16,26,28,43)(18,30,33,45)(20,34)(22,38)(23,29)(24,37)(25,35)(27,42,32,44)(36,40)(39,41)

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, 5712);