#72221466

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

derived

#5464 ⋊ C₂×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#11715753
commutator, hypocenter	#5464	C₂×C₄
Frattini, socle	C₂³	C₂×#1020
Fitting	C₂×#3532	D₃
prime radical	C₂×#3532⋊C₃	C₂
cosocle	C₂×#5464	C₂²
LCS	G ▷ #5464
UCS	C₁ ◁ C₂

1587 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 1587 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.

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

Full subgroup data →

Presentation

Polycyclic presentation:

2^3,3,2^6|a^2=c,e^2=i,f^2=j,d^a=d^2,e^a=efj,e^b=ei,e^d=fi,f^a=fj,f^b=fj,f^d=ef,g^a=h,g^d=h,g^e=gj,g^f=gi,h^a=g,h^d=gh,h^e=hi,h^f=hij,i^a=ij,i^d=j,j^d=ij

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72221473, C₄×#11715753, #72242975, …
maximal normal subgroup of	C₄×#11715753, #72242975, #72243028, …
maximal quotient of	#72221473

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