#72341364

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting

#6372734 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#1020⋊C₂
commutator, hypocenter	#997	C₂³×C₄
Frattini	C₂⁴	C₂³×S₄
Fitting	#6372734	D₃
prime radical	C₂×#11714174	C₂
cosocle	C₂×#997	C₂⁴
hypercenter	C₄⋊C₄	#1020⋊C₂
LCS	G ▷ #997
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

7135 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 64

Polycyclic presentation:

2^5,3,2^5|a^2=e,c^2=k,d^2=e,g^2=k,h^2=k,c^a=ck,c^b=ck,f^d=f^2,g^b=gjk,g^d=hk,g^f=h,h^b=hik,h^d=gk,h^f=ghk,h^g=hk,i^d=j,i^f=ij,i^g=ik,j^d=i,j^f=i,j^h=jk

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#72217027, #72341940
maximal normal subgroup of	C₄×#72217027, #72341940

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