Finite Group Databasebeta

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#72253113

order6912
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC2C22×#11724050
commutator, cosocle, hypocenter#63703C23×C3
Frattini, socleC2×C3C22×C3×#1020
FittingC32×H(7,𝔽2)D3
hypercenterD4#11724050
LCSG ▷ #63703
UCSC1 ◁ C2 ◁ D4

7066 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 7066 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degreeμ(G) = 25
transitive degreeμt(G) = 144

A minimal faithful permutation representation — 3 generators on 25 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

194 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72268997

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