Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C79 ⋊ #13620 |
290 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 111 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)(82,94)(87,110)(93,122)(95,124)(96,98)(97,125)(100,133)(107,127)(109,126)(111,121)(112,139)(114,141)(119,143)(123,129)(128,142)(134,135);(80,82)(81,87)(83,100)(84,96)(85,107)(86,94)(88,114)(89,111)(90,119)(91,121)(92,110)(93,99)(95,103)(97,101)(98,105)(102,134)(104,133)(106,112)(108,127)(109,113)(115,128)(116,142)(117,141)(118,139)(120,143)(122,132)(123,130)(124,138)(125,136)(126,140)(129,131)(135,137);(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)(80,81,84,89,86,92,105,91)(82,93,96,123,94,122,98,129)(83,99,102,130,104,132,137,131)(85,106,101,90,108,118,136,120)(87,109,111,124,110,126,121,95)(88,113,115,138,117,140,116,103)(97,128,127,141,125,142,107,114)(100,119,134,139,133,143,135,112)Polycyclic presentation:
2^8,79|a^2=e,d^2=fg,e^2=h,f^2=gh,g^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,i^a=i^78,c^b=ch,d^b=df,f^b=fg,g^b=gh1768 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 90023);