Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C79×#2135 ⋊ C2 |
| Sylow | C79 ⋊ #13096 |
624 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 624 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 143 points:
(1,2)(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)(80,83)(81,88)(82,94)(84,104)(85,107)(86,101)(87,109)(89,114)(90,117)(91,113)(92,119)(93,99)(95,127)(96,129)(97,126)(98,118)(100,108)(102,112)(103,124)(105,133)(106,125)(110,130)(111,141)(115,131)(116,140)(120,142)(121,139)(122,136)(123,137)(128,143)(132,134)(135,138);(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)(80,82,93,95,86,97,84,103)(81,87,102,121,91,111,89,92)(83,94,99,127,101,126,104,124)(85,106,122,135,108,96,105,128)(88,109,112,139,113,141,114,119)(90,116,134,120,118,110,115,137)(98,130,131,123,117,140,132,142)(100,129,133,143,107,125,136,138);(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)(82,92,97,121)(83,98,101,117)(84,102,93,89)(85,90)(86,91)(87,95)(88,100,113,107)(94,123)(96,120,106,137)(99,132)(103,111)(104,131)(105,134,122,115)(108,118)(109,138,141,143)(110,128)(112,133)(114,136)(116,135)(119,129)(124,130,127,140)(125,139)(126,142)Polycyclic presentation:
2^8,79|a^2=f,b^2=df,e^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,i^a=i^78,d^b=dg,e^b=eh,f^b=fg,g^b=gh,i^b=i^78,d^c=dh,e^c=eh,f^c=fh,i^c=i^78,f^d=fh931 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 87130);