Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2 | C2×#1020 |
| commutator, cosocle, hypocenter | #997 | C22 |
| Fitting | #585 | D3 |
| prime radical | #5950 | C2 |
| hypercenter | C4 | #1020 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C2 ◁ C4 | |
253 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 253 conjugacy classes of subgroups — too many to draw. Only its 16 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,29,20,14,17,13,6,30,5,3,19,31)(2,32,18,11,15,22,10,27,9,7,24,26)(4,12,16,28)(8,21,23,25);(1,2,6,10)(3,11,14,7)(4,24)(5,9,20,18)(8,17)(12,32)(13,26,31,22)(15,16)(19,23)(21,30)(25,29)(27,28)Polycyclic presentation:
2^2,3,2^5|a^2=h,b^2=h,f^2=h,g^2=h,b^a=bh,c^a=c^2,e^a=de,f^a=fg,g^a=gh,d^c=e,e^c=de,f^c=g,g^c=fgh,f^d=fh,f^e=fh,g^e=gh,g^f=gh23 conjugacy classes — view the full character table.
| maximal subgroup of | #11711646, #11711647, #11711648, … |
| maximal normal subgroup of | #11711646, #11711647, #11711648, … |
| maximal quotient of | #11715585, #11715587, #11715590, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20094);