C₁₃×#1738 #13110796

order	1664
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₁₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₃×#1738

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂×D₄
commutator	C₄	C₂×C₄²×C₁₃
Frattini, cosocle	C₂²×C₄	C₂³×C₁₃
socle	C₂²×C₁₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₃	Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂³×C₄×C₁₃ ◁ G

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

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 216 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)(14,15)(16,17)(18,19,21,26,23,27,22,25)(20,28,30,24,29,33,31,32);(15,17)(18,20,22,31,23,29,21,30)(19,24,26,33,27,32,25,28);(20,29)(24,32)(28,33)(30,31)

Conjugacy classes & character table

572 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#15768255, #15768300, #15768363, …
maximal subgroup of	C₁₃×#10744, C₁₃×#10753, C₁₃×#10767, …
maximal normal subgroup of	C₁₃×#10744, C₁₃×#10753, C₁₃×#10767, …
maximal quotient of	C₁₃×#7873, C₁₃×#7876, C₁₃×#7877, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1664, 470);