Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C79 ⋊ #12947 |
380 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 115 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)(81,90)(82,96)(84,103)(85,108)(86,104)(87,111)(88,105)(89,112)(91,101)(92,122)(93,118)(94,120)(95,124)(97,129)(98,134)(99,130)(100,132)(102,106)(107,139)(109,119)(110,131)(113,136)(114,127)(115,142)(116,141)(117,123)(121,137)(125,140)(126,143)(128,135)(133,138);(80,82,86,99,88,100,87,110)(81,89,93,114,94,116,109,115)(83,96,104,130,105,132,111,131)(84,98,106,133,107,135,85,97)(90,112,118,127,120,141,119,142)(91,113,92,95,121,125,123,126)(101,136,122,124,137,140,117,143)(102,138,139,128,108,129,103,134);(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)(80,81,86,93,88,94,87,109)(82,95,99,125,100,126,110,113)(83,101,104,122,105,137,111,117)(84,92,106,121,107,123,85,91)(89,98,114,133,116,135,115,97)(90,102,118,139,120,108,119,103)(96,127,130,141,132,142,131,112)(124,134,140,138,143,128,136,129)Polycyclic presentation:
2^8,79|a^2=f,b^2=f,d^2=gh,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gh,i^a=i^78,d^b=dg,e^b=egh,g^b=gh,d^c=dgh,e^c=egh,g^c=gh1762 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 86244);