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#72229822

order62500
automorphism?
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C1

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#30627468 ⋊ C4

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, hypocenter#30627468C4
FrattiniC53#92563
socleC5#72229821
prime radical, cosocle#72229808C2
LCSG ▷ #30627468
UCSC1

688 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 32 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) = 125
transitive degreeμt(G) = 125

A minimal faithful permutation representation — 3 generators on 125 points:

(2,7,8,20)(3,12,13,36)(4,6,5,19)(9,28,29,68)(10,32,33,54)(11,23,25,60)(14,15,44,47)(16,48,43,39)(17,22,26,59)(18,40,41,82)(21,55,56,81)(24,62,42,88)(27,53,35,50)(30,73,74,111)(31,75,76,99)(34,57,72,105)(37,45,49,46)(38,84,85,52)(51,70,95,116)(58,101,108,86)(61,64,104,92)(63,113,87,120)(65,110,91,69)(66,112,90,107)(67,97,89,119)(71,117,80,106)(77,78,115,103)(79,118,123,109)(83,121,114,96)(93,122,100,98)(94,102,125,124);(1,14,15,36,45,6,18,12,47,41,5,46,40,49,44,4,3,48,39,43,19,16,37,82,13)(2,30,79,124,90,11,34,93,81,87,10,51,83,92,89,17,80,31,77,91,35,9,38,86,42)(7,21,106,69,121,22,58,57,120,118,23,61,28,88,75,50,94,70,119,122,32,78,73,107,84)(8,63,101,76,71,27,66,115,114,95,26,24,64,100,72,33,65,55,123,74,25,67,102,85,29)(20,96,62,117,103,54,52,113,111,125,53,98,110,116,104,60,99,112,68,108,59,109,97,105,56);(1,2,7,20,8)(3,9,21,52,24)(4,17,50,60,33)(5,10,23,53,26)(6,11,22,54,27)(12,38,28,62,55)(13,42,84,56,29)(14,34,61,99,67)(15,31,57,97,64)(16,30,58,98,65)(18,51,94,109,63)(19,35,32,59,25)(36,81,88,68,85)(37,83,106,112,115)(39,86,69,111,100)(40,79,70,113,102)(41,87,118,125,95)(43,91,122,108,74)(44,89,75,104,72)(45,90,121,103,71)(46,80,78,96,66)(47,92,119,105,76)(48,93,73,110,101)(49,77,107,117,114)(82,124,120,116,123)

Polycyclic presentation:

2^2,5^6|a^2=b,e^5=h^3,c^a=c^3,c^b=c^4,d^a=d^2,d^b=d^4,d^c=dg,e^a=e^2h^3,e^b=e^4h^2,e^c=ef,f^a=fh^3,f^b=fh^4,f^d=fh,f^e=fh,g^e=gh,h^a=h^2,h^b=h^4

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

holomorph of5−1+2

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