C₂×C₁₉×#137 #11926250

order	1216
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×C₁₉×#137

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₁₉
prime radical	C₂×#41×C₁₉	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ G

258 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 608

A minimal faithful permutation representation — 4 generators on 37 points:

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

Conjugacy classes & character table

418 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#14564006, #14564110, #14564131, …
maximal subgroup of	C₁₉×#1882, C₁₉×#1902, C₁₉×#1905, …
maximal normal subgroup of	C₁₉×#1882, C₁₉×#1902, C₁₉×#1905, …
maximal quotient of	C₂×C₁₉×#419, C₂×C₁₉×#417, C₂×C₁₉×#428, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1216, 255);