C₁₃×#12264 #15698603

order	3328
nilpotency class	4
isoclinism	isoclinic class of #12148
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₃×#12264

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₁₃
prime radical	C₂⁵×C₁₃	D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂³×C₁₃ ◁ C₂×C₁₃×C₂²⋊C₄ ◁ G

468 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 70 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)(14,15,19,23)(16,25,29,36)(17,30,32,40)(18,33,31,22)(20,24,35,39)(21,34,37,27)(26,38,43,45)(28,41,42,44);(14,16,20,28)(15,21,24,38)(17,26,31,27)(18,34,32,43)(19,29,35,42)(22,36,30,44)(23,37,39,45)(25,40,41,33);(14,17,19,32)(15,22,39,40)(16,26,29,43)(18,20,31,35)(21,36,45,41)(23,33,24,30)(25,38,44,37)(27,42,34,28)

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, 5252);