Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | #392 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #857⋊C2 |
| commutator, cosocle, hypocenter | #95⋊C3 | C22 |
| Frattini, socle | C23 | C2×S4 |
| Fitting | #392 | D3 |
| LCS | G ▷ #95⋊C3 | |
| UCS | C1 ◁ C2 | |
130 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 130 conjugacy classes of subgroups — too many to draw. Only its 12 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,15,23,3,14,22,6,4,20,5,11,13)(2,8,12,7,16,24,10,18,21,9,17,19);(1,2)(3,7)(4,12)(5,9)(6,10)(8,13)(11,19)(14,24)(15,21)(16,20)(17,23)(18,22)Polycyclic presentation:
2^2,3,2^5|b^2=h,d^2=f,e^2=fg,c^a=c^2,d^a=ef,e^a=dfg,f^a=fg,d^b=dfh,e^b=efgh,d^c=efh,e^c=deh,f^c=fg,g^c=fh,e^d=eh21 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72996, #11711934, #11711936, … |
| maximal normal subgroup of | C2×#72996, #11711934, #11711936, … |
| maximal quotient of | #11711089, #11711091, #11711133, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5572);