#72228413

order	2916
automorphism	?
homology	Abelianization (H₁) = C₃³ · Schur multiplier (H₂) = C₂×C₃⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×#6910 ⋊ C₃
Sylow	C₂² ⋊ #10624749

Subgroups

	subgroup	G/H
center	C₃	(C₃×A₄)⋊H(3,𝔽₃)
commutator, cosocle	C₂²×C₃³	C₃³
Frattini	C₃³	C₃²×A₄
Fitting	C₂²×#6910	C₃
socle	C₂²×C₃	C₃²⋊H(3,𝔽₃)
hypercenter	#6910	A₄
hypocenter	C₂²	#10624749
LCS	G ▷ C₂²×C₃³ ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ #6910

556 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 61 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) = 108

A minimal faithful permutation representation — 3 generators on 31 points:

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

Polycyclic presentation:

3^6,2^2|a^3=f^2,b^3=f^2,c^3=f^2,b^a=bd,c^b=cd^2e,d^b=df,d^c=df^2,e^a=ef,e^b=ef^2,e^c=ef,g^c=h,h^c=gh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72260073
maximal subgroup of	A₄×#10624749, #72260073
maximal normal subgroup of	A₄×#10624749, #72260073

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