Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C2 | #1020⋊C2 |
| commutator, prime radical, cosocle, hypocenter | #997 | C22 |
| Frattini | C23 | C2×S4 |
| Fitting | #458 | D3 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C2 | |
112 conjugacy classes of subgroups; only the 11 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 11 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:
(1,12,16,14,4,28,6,15,29,3,17,27)(2,20,23,11,8,31,10,22,32,7,24,30)(5,18,19,13)(9,25,26,21);(1,2,6,10)(3,11,14,7)(4,9,17,26)(5,8,19,24)(12,25,15,21)(13,22,18,20)(16,23,29,32)(27,31,28,30)Polycyclic presentation:
2^2,3,2^5|a^2=h,b^2=h,b^a=bh,c^a=c^2,d^a=e,e^a=d,f^a=fg,d^b=dfgh,e^b=efh,d^c=e,e^c=de,f^c=g,g^c=fg,f^d=fh,g^d=gh,g^e=gh17 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#73029, #11714169, #11714172, … |
| maximal normal subgroup of | C2×#73029, #11714169, #11714172, … |
| maximal quotient of | #11711407, #11711416, #11711425, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5605);