C₂×C₁₃×#454 #13112074

order	1664
nilpotency class	3
isoclinism	isoclinic class of #453
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₃ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×C₁₃×#454

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃	#120
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ G

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

normal subgroups only

This group has 728 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) = 31
transitive degree	μ_t(G) = 416

A minimal faithful permutation representation — 4 generators on 31 points:

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

Conjugacy classes & character table

416 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#15780262, #15780272, #15781946, …
maximal subgroup of	C₁₃×#13055, C₁₃×#13057, C₁₃×#13357, …
maximal normal subgroup of	C₁₃×#13055, C₁₃×#13057, C₁₃×#13357, …
maximal quotient of	C₂×C₁₃×#1597, C₂×C₁₃×#1615, C₂×C₁₃×#1598, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1664, 1748);