Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
82 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 128 points:
(1,65,110,54,72,118,26,58,31,21,55,127,23,61,78,17,32,67,16,121,83,18,24,113,8,19,109,88,28,117,4,89,34,3,95,128,5,111,75,36,6,64,41,107,82,57,76,114)(2,86,120,50,80,124,44,51,47,40,69,125,27,91,98,15,33,94,22,115,100,37,42,105,14,38,119,60,45,123,10,66,48,9,77,126,11,101,97,53,12,93,71,103,99,49,87,106)(7,20,56,108,29,62,79,90,35,68,96,122,73,112,25,59)(13,39,70,104,46,92,85,63,30,84,74,116,81,102,43,52);(1,2,8,14)(3,15,21,53)(4,22,26,71)(5,27,23,11)(6,30,32,13)(7,33,35,12)(9,36,40,17)(10,41,44,16)(18,60,57,50)(19,63,65,52)(20,66,68,51)(24,74,76,70)(25,77,79,69)(28,46,72,81)(29,45,73,80)(31,48,34,47)(37,54,49,88)(38,59,86,90)(39,58,84,89)(42,56,87,96)(43,55,85,95)(61,104,111,116)(62,103,112,115)(64,106,67,105)(75,120,78,119)(82,99,83,100)(91,122,101,108)(92,121,102,107)(93,113,94,114)(97,109,98,110)(117,125,118,126)(123,128,124,127)Polycyclic presentation:
2^4,3,2^3|a^2=h,b^2=cd,c^2=dh,d^2=h,f^2=h,g^2=h,b^a=bc,c^a=cd,d^a=dh,e^a=e^2,f^a=gh,g^a=fh,f^e=g,g^e=fgh,g^f=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #11711668, #11711669, #11711679, … |
| maximal normal subgroup of | #11711668, #11711669, #11711679, … |
| maximal quotient of | #11711471, #11711473, #11711479, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5618);