C₁₃×#1772 #13110830

order	1664
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₁₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₃×#1772

Subgroups

	subgroup	G/H
center	C₂×C₈×C₁₃	D₄
commutator	C₄	C₂×C₄²×C₁₃
Frattini, cosocle	C₂²×C₄	C₂³×C₁₃
socle	C₂²×C₁₃	C₄×D₄
prime radical	C₂³×C₁₃	#41
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₃ ◁ C₂²×C₈×C₁₃ ◁ G

232 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 152 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) = 33
transitive degree	μ_t(G) = 416

A minimal faithful permutation representation — 3 generators on 33 points:

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

Conjugacy classes & character table

728 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#15749859, #15749862, #15749865, …
maximal subgroup of	C₁₃×#7179, C₁₃×#7180, C₁₃×#7181, …
maximal normal subgroup of	C₁₃×#7179, C₁₃×#7180, C₁₃×#7181, …
maximal quotient of	C₁₃×#8266, C₁₃×#8267, C₁₃×#8270, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1664, 504);