Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C256 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C79⋊C256 |
| derived, Sylow | C79 ⋊ C256 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C128 | D79 |
| commutator, hypocenter | C79 | C256 |
| Fitting, cosocle | C79×C128 | C2 |
| socle, prime radical | C2×C79 | C128 |
| LCS | G ▷ C79 | |
| UCS | C1 ◁ C128 | |
18 conjugacy classes of subgroups.
| minimal degree | μ(G) = 335 |
| transitive degree | μt(G) = 20224 |
A minimal faithful permutation representation — 2 generators on 335 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)(80,81,82,89,83,90,96,117,84,91,97,118,102,123,138,173,85,92,98,119,103,124,139,174,107,128,143,178,153,188,208,243,86,93,99,120,104,125,140,175,108,129,144,179,154,189,209,244,111,132,147,182,157,192,212,247,163,198,218,253,228,263,278,299,87,94,100,121,105,126,141,176,109,130,145,180,155,190,210,245,112,133,148,183,158,193,213,248,164,199,219,254,229,264,279,300,114,135,150,185,160,195,215,250,166,201,221,256,231,266,281,302,169,204,224,259,234,269,284,305,238,273,288,309,293,314,320,327,88,95,101,122,106,127,142,177,110,131,146,181,156,191,211,246,113,134,149,184,159,194,214,249,165,200,220,255,230,265,280,301,115,136,151,186,161,196,216,251,167,202,222,257,232,267,282,303,170,205,225,260,235,270,285,306,239,274,289,310,294,315,321,328,116,137,152,187,162,197,217,252,168,203,223,258,233,268,283,304,171,206,226,261,236,271,286,307,240,275,290,311,295,316,322,329,172,207,227,262,237,272,287,308,241,276,291,312,296,317,323,330,242,277,292,313,297,318,324,331,298,319,325,332,326,333,334,335);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)Polycyclic presentation:
2^8,79|a^2=b,b^2=c,c^2=d,d^2=e,e^2=f,f^2=g,g^2=h,i^a=i^785248 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 56093);