Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C2×#510×C79 ⋊ C2 |
| Sylow | C79 ⋊ #13311 |
520 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 520 conjugacy classes of subgroups — too many to draw. Only its 95 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:
(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)(80,83)(81,89)(82,95)(84,101)(85,109)(86,102)(87,103)(88,110)(90,115)(91,120)(92,116)(93,100)(94,121)(96,125)(97,130)(98,126)(99,127)(104,132)(105,134)(106,137)(107,135)(108,136)(111,138)(112,114)(113,131)(117,124)(118,142)(119,133)(122,141)(123,140)(128,139)(129,143);(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)(80,82,87,99)(81,88,92,112)(83,95,103,127)(84,105,108,128)(85,98,86,97)(89,114,116,110)(90,117,119,122)(91,113,93,111)(94,118,123,104)(96,106,129,107)(100,131,120,138)(101,134,136,139)(102,130,109,126)(115,141,133,124)(121,132,140,142)(125,137,143,135);(80,81)(82,94)(83,100)(84,104)(85,93)(86,91)(87,92)(88,96)(89,102)(90,107)(95,124)(97,117)(98,122)(99,123)(101,133)(103,120)(105,111)(106,119)(108,118)(109,116)(110,134)(112,129)(113,128)(114,139)(115,136)(121,126)(125,131)(127,141)(130,140)(132,135)(137,142)(138,143)Polycyclic presentation:
2^8,79|b^2=h,d^2=gh,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,g^a=gh,c^b=cf,d^b=dg,e^b=eh,g^b=gh,e^c=eh,i^c=i^781282 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 88305);