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#72226331

order4608
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24×C3

Aliases

No aliases for this group.

Splittings

FittingD4×H(5,𝔽2) ⋊ C3⋊D3

Subgroups

subgroupG/H
center, Frattini, socleC22C22×A4⋊S4
commutator, cosocleC2×Ω+(4,𝔽3)C23
FittingD4×H(5,𝔽2)C3⋊D3
hypercenterC2×D4A4⋊S4
hypocenterΩ+(4,𝔽3)C2×D4
LCSG ▷ C2×Ω+(4,𝔽3) ▷ Ω+(4,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

3728 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 3728 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 52
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 3 generators on 52 points:

(5,12)(7,41,27,45)(8,24,11,9)(10,25)(14,50,37,51)(15,16,18,29)(19,22,38,20)(21,44,23,39)(26,40,42,43)(28,48,34,52)(31,35,47,33)(32,46,49,36);(2,4)(5,7,19)(6,14,32)(8,21,44)(9,22,38)(10,26,41)(11,27,45)(12,23,40)(13,28,31)(15,34,51)(16,35,49)(17,36,48)(18,37,52)(20,39,25)(24,42,43)(29,46,47)(30,33,50);(1,2,3,4)(5,6)(7,14)(8,16,25,15,9,30)(10,18,24,17,11,29)(12,13)(19,32)(20,34,22,33,21,35)(23,28)(26,37,42,36,27,46)(31,40)(38,50,44,49,39,51)(41,52,43,48,45,47)

Polycyclic presentation:

2^4,3^2,2^5|a^2=d,g^2=k,h^2=k,b^a=bd,c^a=ck,e^c=e^2,f^c=f^2,g^c=hk,g^e=h,g^f=h,h^c=gk,h^e=ghk,h^f=ghk,h^g=hk,i^c=gij,i^e=j,i^f=ghi,i^h=ik,j^c=ghjk,j^e=ij,j^f=gjk,j^g=jk

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

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