Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C79 ⋊ #7423 |
140 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 43 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 — 2 generators on 143 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)(80,82,87,95)(81,88,92,116)(83,100,102,126)(84,107,85,110)(86,113,108,98)(89,117,118,141)(90,120,121,143)(91,122,112,115)(93,123,124,99)(94,103,128,104)(96,101,97,111)(105,139,136,134)(106,140,109,114)(119,130,142,129)(125,132,131,138)(127,137,133,135);(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)(82,93,95,124)(83,99,102,123)(84,106)(85,109)(86,112)(87,92)(88,94)(89,103,118,104)(90,101)(91,108)(96,129,97,130)(98,131,113,125)(100,120)(105,138,136,132)(107,141,110,117)(111,121)(114,139)(115,133)(116,128)(119,137,142,135)(122,127)(126,143)(134,140)Polycyclic presentation:
2^8,79|a^2=d,b^2=h,c^2=fgh,e^2=g,f^2=g,g^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fgh,g^a=gh,i^a=i^78,c^b=cf,d^b=defgh,e^b=egh,f^b=fgh,g^b=gh,d^c=dh1045 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 57138);