C₁₃×#12771 #15699110

order	3328
nilpotency class	4
isoclinism	isoclinic class of #12767
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₃×#12771

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₃	#1790
commutator	C₂×D₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×#41	C₂³×C₁₃
prime radical	C₁₃×#532	C₂²
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂×C₄×C₁₃ ◁ C₂×C₁₃×#41 ◁ G

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

normal subgroups only

This group has 728 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)(14,15,19,25,20,24,34,41)(16,22,27,39,17,28,30,42)(18,23,32,40,31,36,33,43)(21,35,37,29,38,44,45,26);(14,16,20,17)(15,21,24,38)(18,26,31,29)(19,30,34,27)(22,36,28,23)(25,37,41,45)(32,44,33,35)(39,43,42,40);(14,17,20,16)(15,22,24,28)(18,26,31,29)(19,27,34,30)(21,36,38,23)(25,42,41,39)(32,44,33,35)(37,40,45,43)

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, 5759);