Finite Group DatabasebetaFinite Group Databasebeta| order | 20096 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C157 ⋊ Q128 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #23757108 |
| commutator, cosocle | C32×C157 | C22 |
| Frattini | C32 | C2×D157 |
| Fitting | Q64×C157 | C2 |
| socle, prime radical | C2×C157 | D32 |
| hypercenter | Q64 | D157 |
| hypocenter | C157 | Q128 |
| LCS | G ▷ C32×C157 ▷ C16×C157 ▷ C8×C157 ▷ C4×C157 ▷ C2×C157 ▷ C157 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ Q64 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 285 |
| transitive degree | μt(G) = 20096 |
A minimal faithful permutation representation — 2 generators on 285 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)(102,103)(104,105)(106,107)(108,109)(110,111)(112,113)(114,115)(116,117)(118,119)(120,121)(122,123)(124,125)(126,127)(128,129)(130,131)(132,133)(134,135)(136,137)(138,139)(140,141)(142,143)(144,145)(146,147)(148,149)(150,151)(152,153)(154,155)(156,157)(158,160,165,177)(159,166,171,200)(161,179,184,232)(162,186,189,249)(163,191,193,253)(164,195,192,176)(167,202,205,266)(168,207,209,274)(169,211,212,276)(170,213,194,199)(172,214,217,178)(173,201,196,181)(174,206,222,180)(175,210,224,187)(182,239,241,285)(183,243,240,231)(185,197,246,259)(188,251,234,248)(190,198,252,261)(203,268,269,282)(204,270,254,265)(208,275,244,273)(215,279,280,238)(216,228,225,242)(218,267,256,284)(219,237,255,264)(220,277,281,226)(221,235,247,271)(223,278,283,227)(229,263,260,233)(230,262,257,236)(245,272,250,258);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,100,102,104,106,108,110,112,114,116,118,120,122,124,126,128,130,132,134,136,138,140,142,144,146,148,150,152,154,156,157,155,153,151,149,147,145,143,141,139,137,135,133,131,129,127,125,123,121,119,117,115,113,111,109,107,105,103,101,99,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(158,159,165,171)(160,172,177,217)(161,178,184,214)(162,185,189,246)(163,190,193,252)(164,194,192,170)(166,196,200,173)(167,201,205,181)(168,206,209,180)(169,210,212,187)(174,220,222,281)(175,223,224,283)(176,225,195,216)(179,207,232,274)(182,238,241,279)(183,242,240,228)(186,202,249,266)(188,250,234,245)(191,215,253,280)(197,257,259,230)(198,260,261,229)(199,219,213,255)(203,267,269,284)(204,237,254,264)(208,235,244,271)(211,256,276,218)(221,282,247,268)(226,263,277,233)(227,262,278,236)(231,275,243,273)(239,272,285,258)(248,270,251,265)Polycyclic presentation:
2^7,157|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^1561517 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20096, 2727);