Finite Group DatabasebetaFinite Group Databasebeta| order | 12928 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C101 · Schur multiplier (H2) = C22 |
| direct product | C101×#1494 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C101 | C2×D4 |
| commutator | C2×C4 | C22×C4×C101 |
| Frattini, cosocle | C42 | C23×C101 |
| socle | C22×C101 | C2×C22⋊C4 |
| prime radical | #121×C101 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C101 ◁ C2×C42×C101 ◁ G | |
230 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 6464 |
A minimal faithful permutation representation — 3 generators on 125 points:
(1,2,4,7,5,3,8,6)(9,10,12,16,13,15,19,22)(11,14,18,20,17,21,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125);(1,3)(2,5)(4,6)(7,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24);(1,4,5,8)(2,6,3,7)(10,15)(11,17)(16,22)(18,23)4444 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C101×#8348, C101×#8349, C101×#8528, … |
| maximal normal subgroup of | C101×#8348, C101×#8349, C101×#8528, … |
| maximal quotient of | C101×#7654, C101×#7655, C101×#7656, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12928, 226);