Finite Group DatabasebetaFinite Group Databasebeta| order | 4544 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C71 · Schur multiplier (H2) = C24 |
| direct product | #538×C71 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C71 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C71 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C71 ◁ G | |
250 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 87 |
| transitive degree | μt(G) = 1136 |
A minimal faithful permutation representation — 4 generators on 87 points:
(1,2)(3,9)(4,8)(5,7)(6,10)(11,13)(12,15)(14,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87);(1,3,5,11)(2,6,7,14)(4,10,12,16)(8,9,15,13);(2,7)(3,10)(6,13)(8,15)(9,14)(11,16);(2,8)(6,9)(7,15)(13,14)1562 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C71×#1600, C71×#1606, C71×#1612, … |
| maximal normal subgroup of | C71×#1600, C71×#1606, C71×#1612, … |
| maximal quotient of | C71×#2312, C71×#2318, C71×#2321, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4544, 219);