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 | C2 | C2×#1020 |
| commutator, cosocle, hypocenter | C22⋊SL(2,𝔽3) | C22 |
| Fitting | C22×#41 | D3 |
| socle | C23 | C2×S4 |
| prime radical | #5948 | C2 |
| hypercenter | C4 | #1020 |
| LCS | G ▷ C22⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C4 | |
263 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 263 conjugacy classes of subgroups — too many to draw. Only its 17 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 36 points:
(2,3,4)(5,32,24,18,20,17,10,34,9,7,23,35)(6,36,22,15,19,26,14,31,13,11,28,30)(8,16,21,33)(12,25,27,29);(1,4,2,3)(5,6,10,14)(7,15,18,11)(8,19,21,28)(9,22,24,13)(12,23,27,20)(16,31,33,36)(17,26,35,30)(25,32,29,34)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,g^f=gh24 conjugacy classes — view the full character table.
| maximal subgroup of | #11711246, #11711247, #11711252, … |
| maximal normal subgroup of | #11711246, #11711247, #11711252, … |
| maximal quotient of | #11715585, #11715588, #11715592, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20084);