C₁₃×#1750 #13110808

order	1664
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₁₃ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₁₃×#1750

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂³
commutator	C₂²	C₂×C₄²×C₁₃
Frattini, cosocle	C₂²×C₄	C₂³×C₁₃
socle	C₂³×C₁₃	C₂²×C₄
prime radical	C₂⁴×C₁₃	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃ ◁ G

336 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 168 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) = 832

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

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

Conjugacy classes & character table

728 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#15749947, #15749953, #15754439, …
maximal subgroup of	C₁₃×#7214, C₁₃×#7216, C₁₃×#8348, …
maximal normal subgroup of	C₁₃×#7214, C₁₃×#7216, C₁₃×#8348, …
maximal quotient of	C₁₃×#8148, C₁₃×#8149, C₁₃×#8154, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1664, 482);