C₁₃×#12232 #15698571

order	3328
nilpotency class	4
isoclinism	isoclinic class of #12093
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₃×#12232

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₁₃
prime radical	C₂⁴×C₁₃	C₂×D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂³×C₁₃ ◁ C₂×C₁₃×C₂²⋊C₄ ◁ G

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

normal subgroups only

This group has 360 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) = 832

A minimal faithful permutation representation — 3 generators on 45 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,19,22)(16,23,25,28)(17,21,18,26)(20,27,29,24)(30,31)(32,38)(33,36)(34,37)(35,40)(39,44)(41,43)(42,45);(14,16)(15,20)(17,24)(18,27)(19,25)(21,28)(22,29)(23,26)(30,32,34,39)(31,35,37,42)(33,40,41,45)(36,38,43,44);(14,17,19,18)(15,21,22,26)(16,24,25,27)(20,28,29,23)(32,39)(33,41)(35,42)(38,44)

Conjugacy classes & character table

325 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 5220);