#72277162

order	12312
automorphism	?
homology	Abelianization (H₁) = C₂×C₉ · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

Fitting, socle	C₂²×C₃×C₁₉ ⋊ D₃×C₉
derived	C₃×A₄×C₁₉ ⋊ C₂×C₉
Sylow	C₁₉ ⋊ C₉×C₃⋊S₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, hypocenter	C₃×A₄×C₁₉	C₂×C₉
Fitting, socle	C₂²×C₃×C₁₉	D₃×C₉
prime radical	#72277139	C₃
cosocle	C₃×A₄×C₁₉⋊C₃	C₂×C₃
LCS	G ▷ C₃×A₄×C₁₉
UCS	C₁

248 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 35 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) = 26
transitive degree	μ_t(G) = 228

A minimal faithful permutation representation — 3 generators on 26 points:

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

Polycyclic presentation:

2,3^4,19,2^2|b^3=c,d^a=d^2,e^a=e^2,f^a=f^18,f^b=f^17,f^c=f^11,g^a=gh,g^d=h,h^d=gh

Conjugacy classes & character table

93 conjugacy classes — view the full character table.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h");;
gap> coll := SingleCollector(F, [2, 3, 3, 3, 3, 19, 2, 2]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^18);;
gap> SetConjugate(coll, 6, 2, F.6^17);;
gap> SetConjugate(coll, 6, 3, F.6^11);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 4, F.8);;
gap> SetConjugate(coll, 8, 4, F.7*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);