#72277663

order	4374
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃⁴

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

C₃×UT(4,𝔽₃) ⋊ C₂

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle	C₃²×H(3,𝔽₃)	C₂×C₃²
Frattini	C₃³	C₃²×C₃⋊D₃
Fitting	C₃×UT(4,𝔽₃)	C₂
socle	C₃²	(C₃⋊D₃)⋊H(3,𝔽₃)
hypocenter	C₃⁴	C₂×H(3,𝔽₃)
LCS	G ▷ C₃²×H(3,𝔽₃) ▷ C₃⁴
UCS	C₁

1786 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 1786 conjugacy classes of subgroups — too many to draw. Only its 65 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) = 30
transitive degree	μ_t(G) = 81

A minimal faithful permutation representation — 3 generators on 30 points:

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

Polycyclic presentation:

2,3^7|c^b=cd,e^a=e^2,e^b=eg,e^d=eh,f^a=f^2,f^c=fh^2,g^a=g^2,g^c=gh^2,h^a=h^2

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72277681

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, 3, 3, 3]);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 3, F.6*F.8^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 3, F.7*F.8^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);