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#72228630

order15552
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C2×C4

Aliases

No aliases for this group.

Splittings

Fitting, SylowH(5,𝔽3) ⋊ #492

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle#14519105C23
Frattini, socleC3#72213315
Fitting, hypocenterH(5,𝔽3)#492
prime radical#72228607C2
LCSG ▷ #14519105 ▷ #92404 ▷ H(5,𝔽3)
UCSC1

700 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 41 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) = 108
transitive degreeμt(G) = 108

A minimal faithful permutation representation — 3 generators on 108 points:

(1,87,88,19,94,23,90,4)(2,52,8,10,51,59,43,97)(3,17)(5,32,24,20,27,35,29,31)(6,78,33,34,25,92,89,26)(7,106,100,108,101,39,49,37)(9,55)(11,46,93,54,60,53,57,66)(12,18,47,45,103,50,63,28)(13,15,56,75,81,70,74,61)(14,22,79,16,84,71,48,77)(21,85)(30,80,82,68,67,69,73,41)(36,38,62,105,76,44,99,102)(40,58,107,95,72,98,83,42)(65,96);(2,7)(4,19,23,33)(5,24,31,27)(8,42,45,98)(9,49,51,58)(10,39,47,37)(11,62,57,38)(12,40)(13,67,22,61)(14,69,30,71)(15,16,79,77)(18,65,43,101)(20,32,29,91)(21,89,25,87)(26,34,92,88)(28,95,50,100)(36,64,44,53)(41,70,68,82)(46,102)(48,74,73,56)(52,96,63,72)(54,99,60,76)(55,83,103,106)(59,107,97,108)(66,105,93,104)(75,81,84,80)(78,85,90,94);(1,2,3,7)(4,18,22,38)(5,28,30,96)(6,12,17,40)(8,41,44,21)(9,48,39,29)(10,56,58,32)(11,61,65,19)(13,62,23,43)(14,72,31,50)(15,76,78,55)(16,83,85,54)(20,51,73,37)(24,63,71,95)(25,45,68,36)(26,93,81,107)(27,52,69,100)(33,57,67,101)(34,97,84,104)(35,46,86,102)(42,89,53,70)(47,74,49,91)(59,75,105,88)(60,77,106,94)(64,82,98,87)(66,80,108,92)(79,99,90,103)

Polycyclic presentation:

2^6,3^5|a^2=e,b^2=f,c^2=d,d^2=f,b^a=be,c^b=cd,d^b=df,g^a=h,g^b=ghk,g^c=g^2h^2,g^d=h^2k^2,g^e=g^2k^2,g^f=g^2k^2,h^a=g^2k^2,h^b=gh^2k^2,h^c=gh^2,h^d=g,h^e=h^2k^2,h^f=h^2k^2,h^g=hk,i^a=g^2h^2ik^2,i^b=ij,i^c=ij,i^d=i^2j,i^e=hik^2,i^f=i^2,i^g=ik,j^a=gjk,j^b=ij^2,j^c=i,j^d=ij,j^e=ghj,j^f=j^2,j^g=jk,j^h=jk^2,k^c=k^2

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

holomorph of#1088

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