C₁₃×#1605 #13110663

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

Aliases

direct product

C₁₃×#1605

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₁₃
prime radical	C₂³×C₁₃	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ G

238 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 68 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)(18,21)(22,23,26,29)(24,30,32,37)(25,33,35,28)(27,34,36,31);(14,15)(16,18)(17,19)(20,21)(22,24,26,32)(23,27,29,36)(25,34,35,31)(28,37,33,30);(14,16,17,20)(15,18,19,21)

Conjugacy classes & character table

299 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#15775736, #15775746, #15775753, …
maximal subgroup of	C₁₃×#12220, C₁₃×#12222, C₁₃×#12223, …
maximal normal subgroup of	C₁₃×#12220, C₁₃×#12222, C₁₃×#12223, …
maximal quotient of	C₁₃×#7757, C₁₃×#7761, C₁₃×#7763, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1664, 337);