C₂×C₁₉×#123 #11926200

order	1216
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₉ · Schur multiplier (H₂) = C₂⁵

Aliases

C₂×C₁₉×#123

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₁₉
prime radical	C₂²×D₄×C₁₉	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ G

342 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 178 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 33
transitive degree	μ_t(G) = 608

A minimal faithful permutation representation — 4 generators on 33 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1216, 205);