C₁₃×#1609 #13110667

order	1664
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₃×#1609

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₁₃
prime radical	C₁₃×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ G

216 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 37
transitive degree	μ_t(G) = 416

A minimal faithful permutation representation — 3 generators on 37 points:

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

Conjugacy classes & character table

299 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₁₃×#1609, #15835073, #15835226, …
maximal subgroup of	C₂×C₁₃×#1609, C₁₃×#18107, C₁₃×#18126, …
maximal normal subgroup of	C₂×C₁₃×#1609, C₁₃×#18107, C₁₃×#18126, …
maximal quotient of	C₁₃×#7788, C₁₃×#7789, C₁₃×#7792, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1664, 341);