C₁₃×#3415 #13112473

order	1664
nilpotency class	4
isoclinism	isoclinic class of #3415
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₃×#3415

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₈	C₂⁴×C₁₃
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₁₃×#41 ◁ C₁₃×#131 ◁ G

370 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 180 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) = 45
transitive degree	μ_t(G) = 416

A minimal faithful permutation representation — 4 generators on 45 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,21)(17,20)(18,31)(19,24)(22,28)(23,29)(25,35)(26,32)(27,41)(30,34)(33,42)(36,40)(37,39)(38,43)(44,45);(15,20)(17,28)(18,29)(21,32)(22,23)(24,34)(26,40)(27,39)(30,42)(31,33)(35,43)(36,37)(38,45)(41,44);(14,16)(15,21)(17,26)(18,27)(19,25)(20,32)(22,36)(23,37)(24,35)(28,40)(29,39)(30,38)(31,41)(33,44)(34,43)(42,45);(16,25)(21,35)(26,38)(27,39)(32,43)(36,44)(37,41)(40,45)

Conjugacy classes & character table

416 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#15783603, #15783617, #16019665, …
maximal subgroup of	C₁₃×#13669, C₁₃×#13671, C₁₃×#33854, …
maximal normal subgroup of	C₁₃×#13669, C₁₃×#13671, C₁₃×#33854, …
maximal quotient of	C₁₃×#33381, C₁₃×#33410, C₁₃×#33454, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1664, 2147);