Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C79 ⋊ #13078 |
454 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 454 conjugacy classes of subgroups — too many to draw. Only its 131 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:
(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(80,83)(81,89)(82,94)(84,104)(85,109)(86,100)(87,101)(88,113)(90,121)(91,98)(92,119)(93,122)(95,129)(96,126)(97,128)(99,107)(102,132)(103,125)(105,133)(106,136)(108,127)(110,118)(111,137)(112,117)(114,141)(115,140)(116,131)(120,139)(123,143)(124,142)(130,138)(134,135);(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)(80,82,84,103,86,97,106,111,87,96,107,95,85,108,105,130)(81,88,112,93,92,115,134,124,91,116,90,120,110,114,102,123)(83,94,104,125,100,128,136,137,101,126,99,129,109,127,133,138)(89,113,117,122,119,140,135,142,98,131,121,139,118,141,132,143);(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)(80,81,85,92,87,91,86,110)(82,93,97,123,96,120,108,124)(83,98,100,119,101,89,109,118)(84,102,105,112,107,134,106,90)(88,111,114,95,116,130,115,103)(94,122,127,142,126,139,128,143)(99,132,133,121,104,135,136,117)(113,138,140,129,131,137,141,125)Polycyclic presentation:
2^8,79|a^2=f,b^2=dfg,d^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,i^a=i^78,d^b=dh,e^b=eh,f^b=fh,i^b=i^78,d^c=dh,f^c=fh1294 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 87038);