Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C79 ⋊ #9336 |
360 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 91 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) = 10112 |
A minimal faithful permutation representation — 3 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)(80,83,85,95)(81,87,88,94)(82,91,92,106)(84,98,90,96)(86,99,101,109)(89,105,103,102)(93,108,97,107)(100,111,104,110)(112,115)(113,120)(114,124)(116,128)(117,132)(118,129)(119,133)(121,136)(122,137)(123,138)(125,140)(126,142)(127,131)(130,141)(134,143)(135,139);(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)(80,82,90,93,85,92,84,97)(81,86,89,104,88,101,103,100)(83,91,96,108,95,106,98,107)(87,99,105,110,94,109,102,111)(112,114,116,130,118,126,117,125)(113,119,121,134,122,135,131,123)(115,124,128,141,129,142,132,140)(120,133,136,143,137,139,127,138);(80,81,84,89,85,88,90,103)(82,86,93,100,92,101,97,104)(83,94,96,105,95,87,98,102)(91,99,107,110,106,109,108,111)(112,113)(114,123,126,134)(115,127)(116,121)(117,131)(118,122)(119,125,135,130)(120,128)(124,139,142,133)(129,136)(132,137)(138,140,143,141)Polycyclic presentation:
2^8,79|a^2=f,b^2=dfg,c^2=g,d^2=gh,e^2=gh,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,e^b=eg,f^b=fh,i^b=i^78,d^c=dg,e^c=egh,f^c=fg1282 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 66552);