Finite Group DatabasebetaFinite Group Databasebeta| order | 12928 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C101 ⋊ SD128 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #20265012 |
| commutator, cosocle | C32×C101 | C22 |
| Frattini | C32 | C2×D101 |
| Fitting, prime radical | D32×C101 | C2 |
| socle | C2×C101 | D32 |
| hypercenter | D32 | D101 |
| hypocenter | C101 | SD128 |
| LCS | G ▷ C32×C101 ▷ C16×C101 ▷ C8×C101 ▷ C4×C101 ▷ C2×C101 ▷ C101 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ D32 | |
38 conjugacy classes of subgroups.
| minimal degree | μ(G) = 165 |
| transitive degree | μt(G) = 6464 |
A minimal faithful permutation representation — 2 generators on 165 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)(103,109)(104,115)(105,116)(106,121)(107,125)(110,111)(112,140)(113,142)(114,131)(117,148)(118,151)(119,132)(120,128)(122,156)(123,136)(124,129)(126,139)(127,130)(133,153)(134,152)(135,138)(137,158)(141,165)(143,144)(145,149)(146,147)(150,164)(154,161)(155,160)(157,163)(159,162);(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)(102,103,108,114)(104,109,119,131)(105,120,123,154)(106,124,126,157)(107,127,125,113)(110,132,135,115)(111,136,138,116)(112,139,141,121)(117,140,150,165)(118,142,149,130)(122,155,146,153)(128,159,161,144)(129,162,163,143)(133,164,160,148)(134,145,158,151)(137,147,152,156)Polycyclic presentation:
2^7,101|a^2=g,c^2=de,d^2=ef,e^2=fg,f^2=g,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,h^a=h^100,c^b=cd,d^b=de,e^b=ef,f^b=fg985 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12928, 2725);