Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C79 ⋊ #13743 |
126 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 126 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 207 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 3 generators on 207 points:
(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(81,89)(88,118)(90,122)(91,123)(92,124)(93,116)(94,133)(100,132)(107,166)(112,174)(119,178)(120,179)(121,131)(125,181)(126,182)(127,176)(128,183)(129,167)(130,184)(134,196)(135,147)(140,202)(144,204)(148,194)(149,195)(150,164)(160,197)(168,172)(180,190)(185,189)(187,203)(191,193);(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)(80,82,87,99)(81,88,89,118)(83,101,106,154)(84,108,111,171)(85,113,115,175)(86,117,114,98)(90,125,122,181)(91,128,123,183)(92,130,124,184)(93,131,116,121)(94,132,133,100)(95,136,139,103)(96,141,143,102)(97,145,146,109)(104,161,163,207)(105,165,162,153)(107,119,166,178)(110,173,156,170)(112,120,174,179)(126,187,182,203)(127,189,176,185)(129,193,167,191)(134,197,196,160)(135,150,147,164)(137,200,201,186)(138,159,177,188)(140,194,202,148)(142,157,169,192)(144,195,204,149)(151,199,205,155)(152,198,206,158)(168,190,172,180);(80,81)(82,94)(83,100)(84,107)(85,112)(86,116)(87,89)(88,95)(90,103)(91,102)(92,109)(93,114)(96,140)(97,144)(98,147)(99,133)(101,128)(104,160)(105,164)(106,132)(108,125)(110,172)(111,166)(113,134)(115,174)(117,135)(118,139)(119,152)(120,151)(121,177)(122,136)(123,141)(124,145)(126,186)(127,188)(129,192)(130,137)(131,138)(142,203)(143,202)(146,204)(148,155)(149,158)(150,162)(153,193)(154,183)(156,168)(157,167)(159,176)(161,190)(163,197)(165,191)(169,187)(170,189)(171,181)(173,185)(175,196)(178,206)(179,205)(180,207)(182,200)(184,201)(194,199)(195,198)Polycyclic presentation:
2^8,79|b^2=h,d^2=ef,e^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ch,d^a=de,e^a=ef,f^a=fg,g^a=gh,d^b=de,e^b=ef,f^b=fg,g^b=gh,i^b=i^781528 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 90630);