Finite Group DatabasebetaFinite Group Databasebeta| order | 12928 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | #532×C101 ⋊ C2 |
| Sylow | C101 ⋊ #3529 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D101 |
| commutator, socle, cosocle | C22×C101 | C25 |
| Frattini | C22 | C24×D101 |
| Fitting | #532×C101 | C2 |
| hypercenter | #532 | D101 |
| hypocenter | C101 | #3529 |
| LCS | G ▷ C22×C101 ▷ C101 | |
| UCS | C1 ◁ C2×C4 ◁ #532 | |
1058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 3232 |
A minimal faithful permutation representation — 5 generators on 125 points:
(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)(110,113,114,121)(111,117,116,123)(112,120,119,125)(115,122,118,124);(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)(103,107)(106,109)(111,116)(112,119)(117,123)(120,125);(103,107)(104,108)(112,119)(115,118)(120,125)(122,124);(102,104)(103,106)(105,108)(107,109)(110,112)(111,115)(113,120)(114,119)(116,118)(117,122)(121,125)(123,124);(102,103)(104,106)(105,107)(108,109)(110,111)(112,118)(113,117)(114,116)(115,119)(120,124)(121,123)(122,125)Polycyclic presentation:
2^7,101|e^2=f,b^a=bf,c^a=cg,d^a=dfg,c^b=cfg,d^b=df,h^d=h^1001438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12928, 18646);