#72256496

order	12288
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Splittings

Fitting

#72219291 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₂²	C₂×#72209642
commutator, cosocle	C₂²×C₂²⋊(C₂²⋊A₄)	C₂⁴
Frattini, socle	C₂⁶	C₂³×S₄
Fitting	#72219291	D₃
hypercenter	C₂³	#72209642
hypocenter	C₂²⋊(C₂²⋊A₄)	C₂×#120
LCS	G ▷ C₂²×C₂²⋊(C₂²⋊A₄) ▷ C₂²⋊(C₂²⋊A₄)
UCS	C₁ ◁ C₂² ◁ C₂³

Only the 196 normal subgroups are shown; non-normal subgroups are omitted.

normal subgroups only

Only the group's 196 normal subgroups are shown; non-normal subgroups are omitted.

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

Full subgroup data →

Presentation

Polycyclic presentation:

2^6,3,2^6|c^a=cf,c^b=ce,g^d=g^2,h^a=hj,h^b=hl,h^d=hi,h^g=i,i^a=ik,i^b=im,i^g=hi,j^d=jk,j^g=k,k^g=jk,l^d=lm,l^g=m,m^g=lm

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

Relationships

automorphism group of

C₂×C₄×Q₈

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", "l", "m");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2]);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 7, 4, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.12);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 8, 7, F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.13);;
gap> SetConjugate(coll, 9, 7, F.8*F.9);;
gap> SetConjugate(coll, 10, 4, F.10*F.11);;
gap> SetConjugate(coll, 10, 7, F.11);;
gap> SetConjugate(coll, 11, 7, F.10*F.11);;
gap> SetConjugate(coll, 12, 4, F.12*F.13);;
gap> SetConjugate(coll, 12, 7, F.13);;
gap> SetConjugate(coll, 13, 7, F.12*F.13);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);