Finite Group DatabasebetaFinite Group Databasebeta| order | 12928 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C101 ⋊ Q128 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #20265012 |
| commutator, cosocle | C32×C101 | C22 |
| Frattini | C32 | C2×D101 |
| Fitting | Q64×C101 | C2 |
| socle, prime radical | C2×C101 | D32 |
| hypercenter | Q64 | D101 |
| hypocenter | C101 | Q128 |
| LCS | G ▷ C32×C101 ▷ C16×C101 ▷ C8×C101 ▷ C4×C101 ▷ C2×C101 ▷ C101 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ Q64 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 229 |
| transitive degree | μt(G) = 12928 |
A minimal faithful permutation representation — 2 generators on 229 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)(102,104,109,121)(103,110,115,144)(105,123,128,176)(106,130,133,193)(107,135,137,197)(108,139,136,120)(111,146,149,210)(112,151,153,218)(113,155,156,220)(114,157,138,143)(116,158,161,122)(117,145,140,125)(118,150,166,124)(119,154,168,131)(126,183,185,229)(127,187,184,175)(129,141,190,203)(132,195,178,192)(134,142,196,205)(147,212,213,226)(148,214,198,209)(152,219,188,217)(159,223,224,182)(160,172,169,186)(162,211,200,228)(163,181,199,208)(164,221,225,170)(165,179,191,215)(167,222,227,171)(173,207,204,177)(174,206,201,180)(189,216,194,202);(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)(102,103,109,115)(104,116,121,161)(105,122,128,158)(106,129,133,190)(107,134,137,196)(108,138,136,114)(110,140,144,117)(111,145,149,125)(112,150,153,124)(113,154,156,131)(118,164,166,225)(119,167,168,227)(120,169,139,160)(123,151,176,218)(126,182,185,223)(127,186,184,172)(130,146,193,210)(132,194,178,189)(135,159,197,224)(141,201,203,174)(142,204,205,173)(143,163,157,199)(147,211,213,228)(148,181,198,208)(152,179,188,215)(155,200,220,162)(165,226,191,212)(170,207,221,177)(171,206,222,180)(175,219,187,217)(183,216,229,202)(192,214,195,209)Polycyclic presentation:
2^7,101|a^2=g,b^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,c^b=cd,d^b=de,e^b=ef,f^b=fg,h^b=h^100985 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12928, 2727);