Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#857 |
| commutator, prime radical, cosocle, hypocenter | #95⋊C3 | C22 |
| Frattini, socle | C23 | C2×S4 |
| Fitting | #376 | D3 |
| hypercenter | C4 | #857 |
| LCS | G ▷ #95⋊C3 | |
| UCS | C1 ◁ C2 ◁ C4 | |
104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 44 points:
(1,2,12)(3,8,6)(4,9,11)(5,7,10)(13,40,28,26,29,41,18,24,31,15,16,39)(14,37,35,23,30,44,22,33,36,19,20,43)(17,25,32,42)(21,34,27,38);(2,6)(7,11)(8,10)(9,12)(13,14,18,22)(15,23,26,19)(16,27,29,21)(17,30,32,20)(24,34,40,38)(25,33,42,37)(28,36,31,35)(39,43,41,44)Polycyclic presentation:
2^2,3,2^5|a^2=h,b^2=h,d^2=f,e^2=fg,b^a=bh,c^a=c^2,d^a=ef,e^a=dfg,f^a=fg,d^c=efh,e^c=deh,f^c=fg,g^c=fh,e^d=eh24 conjugacy classes — view the full character table.
| maximal subgroup of | #11711208, #11711209, #11711227, … |
| maximal normal subgroup of | #11711208, #11711209, #11711227, … |
| maximal quotient of | #11710980, #11711006, #11711162, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5562);