Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
140 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 26 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2,3,4)(5,25,28,11,16,22,7,20,26,6,19,27)(8,21,17,10,14,18,13,12,23,15,9,24);(5,6,8,10,7,11,13,15)(9,17,16,26,14,23,19,28)(12,22,20,24,21,27,25,18)Polycyclic presentation:
2^4,3,2^3|a^2=d,b^2=c,d^2=h,f^2=h,g^2=h,b^a=bdh,e^a=e^2,f^a=gh,g^a=fh,d^b=dh,f^e=g,g^e=fgh,g^f=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#73076, #11712685, #11712686, … |
| maximal normal subgroup of | C2×#73076, #11712685, #11712686, … |
| maximal quotient of | #11711083, #11711094, #11711095, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5652);