#72343706

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting

#392141 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂²	#11713200
commutator	#5950	C₂²×C₄
Frattini	C₂³×C₄	C₂²×S₄
Fitting	#392141	D₃
prime radical	C₂×#11715828	C₂
cosocle	C₂×#5950	C₂³
hypercenter	C₂×D₄	#1020⋊C₂
hypocenter	#997	C₂×C₂²⋊C₄
LCS	G ▷ #5950 ▷ #997
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₄

7873 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 7873 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.

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

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 64

Polycyclic presentation:

2^5,3,2^5|c^2=e,d^2=k,e^2=k,c^a=ck,c^b=cd,d^b=dk,e^b=ek,f^a=f^2,f^c=f^2,g^a=ghik,g^c=gh,g^f=h,h^a=hjk,h^f=gh,i^a=ij,i^c=ij,i^f=ij,i^g=ik,i^h=ik,j^f=i,j^g=jk

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343749, #72343956, #72343985
maximal normal subgroup of	#72343749, #72343956, #72343985

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", "i", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.11);;
gap> SetPower(coll, 5, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 2, F.4*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 3, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.9*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 6, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.11);;
gap> SetConjugate(coll, 9, 8, F.9*F.11);;
gap> SetConjugate(coll, 10, 6, F.9);;
gap> SetConjugate(coll, 10, 7, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);