#72241002

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting

#6628883 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#11713200
commutator, cosocle	#5950	C₂⁴
Frattini	C₂²×C₄	C₂³×S₄
Fitting	#6628883	D₃
hypercenter	C₂×D₄	#1020⋊C₂
hypocenter	#997	C₂²×D₄
LCS	G ▷ #5950 ▷ #997
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₄

At least 6025 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has at least 6025 conjugacy classes of subgroups — too many to draw. Only its 125 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

(1,25,13,20,19,11,3,31,17,2,30,22,6,26,24,7,14,18,9,32,5,8,27,29)(4,15,23,21,16,10,12,28);(4,14)(5,17)(10,25)(11,18)(12,27)(13,24)(15,26)(16,19)(21,31)(22,29)(23,30)(28,32);(2,8)(7,20)(10,15)(11,18)(21,28)(22,29)(25,26)(31,32);(2,7)(3,9)(8,20)(10,21)(11,22)(12,23)(13,24)(15,28)(18,29)(25,31)(26,32)(27,30)

Polycyclic presentation:

2^5,3,2^5|d^2=ek,e^2=k,g^2=k,h^2=k,d^a=de,d^b=dk,e^a=ek,f^c=f^2,g^c=gh,g^d=gik,g^f=h,h^c=hk,h^d=hijk,h^f=ghk,h^g=hk,i^c=j,i^f=ij,i^h=ik,j^c=i,j^f=i,j^g=jk,j^h=jk

Conjugacy classes & character table

79 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72245616, C₂×#72241002
maximal normal subgroup of	#72245616, C₂×#72241002
maximal quotient of	C₂×#72241002, #72343749

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