C₁₃×#12224 #15698563

order	3328
nilpotency class	4
isoclinism	isoclinic class of #12099
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₃×#12224

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₁₃
prime radical	C₂³×C₁₃	#120
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂³×C₁₃ ◁ C₂×C₁₃×C₄⋊C₄ ◁ G

380 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 74 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)(16,28,31,41)(17,33,35,44)(18,37,29,23)(20,27,40,26)(21,36,43,30)(22,39,42,34)(24,38,45,32);(14,16,19,31)(15,21,25,43)(17,34,35,39)(18,36,29,30)(20,32,40,38)(22,45,42,24)(23,41,37,28)(26,33,27,44);(14,17,20,36)(15,22,27,23)(16,29,32,34)(18,38,39,31)(19,35,40,30)(21,41,44,45)(24,43,28,33)(25,42,26,37)

Conjugacy classes & character table

364 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 5212);