Finite Group DatabasebetaFinite Group Databasebeta| order | 4544 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C71 · Schur multiplier (H2) = C22 |
| direct product | #471×C71 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C71 | #121 |
| commutator, Frattini, cosocle | C2×C4 | C23×C71 |
| prime radical | AGL(1,ℤ/8)×C71 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C71 ◁ C2×C4×C71 ◁ G | |
104 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 104 conjugacy classes of subgroups — too many to draw. Only its 50 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 — 3 generators on 87 points:
(2,6)(3,9)(4,12)(8,14)(10,11)(13,15)(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);(1,2,4,6,5,8,12,14)(3,7,10,13,11,16,9,15);(1,3)(2,7)(4,10)(5,11)(6,13)(8,16)(9,12)(14,15)1136 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×#471×C71, C71×#3067, C71×#3068, … |
| maximal normal subgroup of | C2×#471×C71, C71×#3067, C71×#3068, … |
| maximal quotient of | C71×#1665, C71×#1666, C71×#1671, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4544, 152);