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#72226037

order2304
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C22×C3

Aliases

No aliases for this group.

Splittings

FittingC4×H(5,𝔽2) ⋊ C3⋊D3
derivedΩ+(4,𝔽3) ⋊ C2×C4

Subgroups

subgroupG/H
center, Frattini, socleC22C2×A4⋊S4
commutator, hypocenterΩ+(4,𝔽3)C2×C4
FittingC4×H(5,𝔽2)C3⋊D3
prime radicalC2×SO+(4,𝔽3)C2
cosocleC2×Ω+(4,𝔽3)C22
hypercenterC2×C4A4⋊S4
LCSG ▷ Ω+(4,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

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

normal subgroups only

This group has 738 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) = 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);(1,2,3,4)(5,14,19,6,7,32)(8,34,44,15,21,51)(9,35,38,16,22,49)(10,36,41,17,26,48)(11,37,45,18,27,52)(12,28,40,13,23,31)(20,50,25,33,39,30)(24,46,43,29,42,47);(8,9,25)(10,11,24)(15,16,30)(17,18,29)(20,21,22)(26,27,42)(33,34,35)(36,37,46)(38,39,44)(41,45,43)(47,48,52)(49,50,51)

Polycyclic presentation:

2^3,3^2,2^5|a^2=c,f^2=j,g^2=j,b^a=bj,d^b=d^2,e^b=e^2,f^b=gj,f^d=g,f^e=g,g^b=fj,g^d=fgj,g^e=fgj,g^f=gj,h^b=fhi,h^d=i,h^e=fgh,h^g=hj,i^b=fgij,i^d=hi,i^e=fij,i^f=ij

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72226133, #72226331, #72226339
maximal normal subgroup of#72226133, #72226331, #72226339

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