C₁₃×#12068 #15698407

order	3328
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₃×#12068

Subgroups

	subgroup	G/H
center	C₄×C₁₃	#442
commutator	C₈	C₂²×C₈×C₁₃
Frattini, cosocle	C₄×C₈	C₂³×C₁₃
socle	C₂×C₁₃	#1580
prime radical	C₁₃×#508	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃ ◁ C₂×C₈×C₁₃ ◁ C₁₃×#346 ◁ G

204 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 102 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)(14,15,19,24,20,25,30,36,21,26,18,29,35,16,27,23)(17,28,31,42,32,22,37,39,33,40,34,38,45,41,43,44);(14,16)(15,20)(17,28)(18,23)(19,36)(21,25)(22,32)(24,27)(26,35)(29,30)(31,38)(33,40)(34,42)(37,44)(39,43)(41,45);(14,17,30,37,35,45,19,31,21,33,27,43,20,32,18,34)(15,22,23,42,16,28,29,44,26,41,36,38,25,40,24,39)

Conjugacy classes & character table

598 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3328, 5056);