Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
151 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 151 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,41,49,14,5,43,21,39,26,3,48,62,7,16,47,38,24,44,4,59,23,15,19,61)(2,36,56,28,10,53,18,51,34,8,45,64,12,29,55,35,25,54,9,57,33,13,31,63)(6,17,20,40,27,42,50,60)(11,30,32,52,22,37,46,58);(1,2,7,12)(3,13,15,8)(4,18,21,9)(5,22,24,11)(6,25,27,10)(14,28,38,35)(16,30,41,37)(17,29,42,36)(19,32,48,46)(20,31,50,45)(23,34,26,33)(39,58,59,52)(40,57,60,51)(43,53,44,54)(47,55,49,56)(61,64,62,63)Polycyclic presentation:
2^4,3,2^3|a^2=h,b^2=c,c^2=h,f^2=h,g^2=h,b^a=bd,c^a=ch,e^a=e^2,f^a=gh,g^a=fh,d^b=dh,f^e=g,g^e=fgh,g^f=gh31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#73130, #11712759, #11712950, … |
| maximal normal subgroup of | C2×#73130, #11712759, #11712950, … |
| maximal quotient of | #11711148, #11711166, #11711170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5706);