Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C79 ⋊ #13608 |
308 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 308 conjugacy classes of subgroups — too many to draw. Only its 67 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)(80,83,86,99)(81,88,91,114)(82,93,96,123)(84,101,102,97)(85,98,106,132)(87,107,110,111)(89,116,117,112)(90,113,104,130)(92,118,115,137)(94,124,127,121)(95,122,105,131)(100,133,134,128)(103,135,136,129)(108,138,109,139)(119,140,120,141)(125,143,126,142);(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)(80,82,86,96)(81,87,91,110)(83,93,99,123)(84,94,102,127)(85,105,106,95)(88,111,114,107)(89,115,117,92)(90,108,104,109)(97,121,101,124)(98,131,132,122)(100,120,134,119)(103,125,136,126)(112,137,116,118)(113,139,130,138)(128,141,133,140)(129,142,135,143);(80,81)(82,92,95,119)(83,88)(84,100,102,134)(85,104,106,90)(86,91)(87,94,108,125)(89,103)(93,118,122,140)(96,115,105,120)(97,128,101,133)(98,130,132,113)(99,114)(107,124,138,143)(109,126,110,127)(111,121,139,142)(112,129)(116,135)(117,136)(123,137,131,141)Polycyclic presentation:
2^8,79|a^2=e,b^2=h,c^2=h,d^2=g,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,c^b=cg,d^b=dg,e^b=efgh,f^b=fh,i^b=i^78,d^c=dh,e^d=egh,f^e=fh922 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 89967);