Finite Group DatabasebetaFinite Group Databasebeta| order | 4096 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C162 · Schur multiplier (H2) = C2×C162 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C16 | C82 |
| commutator | C8 | C2×C162 |
| Frattini, cosocle | #94852 | C23 |
| socle, prime radical | C23 | #96658 |
| LCS | G ▷ C8 ▷ C1 | |
| UCS | C1 ◁ C22×C16 ◁ G | |
| LECS | G ▷ #94852 ▷ C43 ▷ C23 ▷ C1 | |
1299 conjugacy classes of subgroups; only the 643 normal subgroups are shown.
This group has 1299 conjugacy classes of subgroups — too many to draw. Only its 643 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 160 |
| transitive degree | μt(G) = 4096 |
A minimal faithful permutation representation — 3 generators on 160 points:
(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(33,34,37,43)(35,47,49,82,51,85,90,134)(36,42,53,69)(38,45,59,75)(39,44,60,77)(40,46,58,76)(41,64,66,109,68,110,116,94)(48,81,86,129,88,132,136,158)(50,84,91,115,93,67,111,73)(52,83,57,102,92,135,103,133)(54,71,97,119)(55,70,98,121)(56,72,96,120)(61,79,105,74)(62,78,106,127)(63,80,104,126)(65,108,112,148,114,149,152,140)(87,131,137,151,139,113,150,117)(89,130,95,141,138,159,142,157)(99,123,144,118)(100,122,145,155)(101,124,143,154)(107,128,147,125)(146,156,160,153);(17,18,19,22,20,23,25,28,21,24,26,29,27,30,31,32)(33,35)(34,41,44,67,46,68,79,84)(36,48)(37,57,58,103)(38,50)(39,52)(40,51)(42,65,70,113,72,114,123,131)(43,73,74,66,76,115,77,116)(45,47,78,83,80,85,128,135)(49,60,90,105)(53,95,96,142)(54,87)(55,89)(56,88)(59,64,104,110)(61,92)(62,94)(63,93)(69,117,118,112,120,151,121,152)(71,81,122,130,124,132,156,159)(75,102,125,82,126,133,127,134)(86,98,136,144)(91,106,111,147)(97,108,143,149)(99,138)(100,140)(101,139)(107,109)(119,141,153,129,154,157,155,158)(137,145,150,160)(146,148);(33,36,38,54,61,99,107,146,40,56,63,101,39,55,62,100)(34,42,45,71,79,123,128,156,46,72,80,124,44,70,78,122)(35,48,50,87,92,138,109,148,51,88,93,139,52,89,94,140)(37,53,59,97,105,144,147,160,58,96,104,143,60,98,106,145)(41,65,47,81,84,131,135,159,68,114,85,132,67,113,83,130)(43,69,75,119,74,118,125,153,76,120,126,154,77,121,127,155)(49,86,91,137,103,142,110,149,90,136,111,150,57,95,64,108)(66,112,82,129,115,151,133,157,116,152,134,158,73,117,102,141)Polycyclic presentation:
2^12|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,f^2=il,g^2=j,h^2=k,i^2=l,b^a=bf,d^b=di,e^a=eil,e^d=el,g^b=gl,h^a=hl736 conjugacy classes — view the full character table.
| maximal subgroup of | #72234276, #72234314 |
| maximal normal subgroup of | #72234276, #72234314 |
| maximal quotient of | #72234350, #72234359 |
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", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9*F.12);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.11);;
gap> SetPower(coll, 9, F.12);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.12);;
gap> SetConjugate(coll, 5, 4, F.5*F.12);;
gap> SetConjugate(coll, 7, 2, F.7*F.12);;
gap> SetConjugate(coll, 8, 1, F.8*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);