Finite Group DatabasebetaFinite Group Databasebeta| order | 12928 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | #585×D101 |
| Fitting | #585×C101 ⋊ C2 |
| Sylow | C101 ⋊ C2×#585 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C24×D101 |
| commutator, socle, cosocle | C2×C101 | C26 |
| Frattini | C2 | C25×D101 |
| Fitting | #585×C101 | C2 |
| hypercenter | #585 | D101 |
| hypocenter | C101 | C2×#585 |
| LCS | G ▷ C2×C101 ▷ C101 | |
| UCS | C1 ◁ C4 ◁ #585 | |
5956 conjugacy classes of subgroups; only the 3203 normal subgroups are shown.
This group has 5956 conjugacy classes of subgroups — too many to draw. Only its 3203 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 117 |
| transitive degree | μt(G) = 1616 |
A minimal faithful permutation representation — 6 generators on 117 points:
(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(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)(88,89)(90,91)(92,93)(94,95)(96,97)(98,99)(100,101);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,100,101,99,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(102,105,106,113)(103,109,108,115)(104,112,111,117)(107,114,110,116);(103,108)(107,110)(109,115)(114,116);(104,111)(107,110)(112,117)(114,116);(102,104)(103,107)(105,112)(106,111)(108,110)(109,114)(113,117)(115,116);(102,103)(104,110)(105,109)(106,108)(107,111)(112,116)(113,115)(114,117)1768 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12928, 19316);