Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | C79 ⋊ #13724 |
174 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 174 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 151 |
| transitive degree | μt(G) = 20224 |
A minimal faithful permutation representation — 3 generators on 151 points:
(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)(81,85)(84,87)(88,91,92,109,112,138,116,107,95,110,113,141,119,135,104,108,96,111,114,143,133,148,136,140,118,137,115,106,94,105,93,90)(89,98,99,126,117,139,132,124,102,127,129,149,134,147,121,125,103,128,130,150,146,151,145,144,120,142,131,123,101,122,100,97);(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)(80,82,83,86)(81,84,85,87)(88,90,104,135,94,106,113,110,118,140,116,138,133,143,92,91,96,108,93,105,119,141,115,137,95,107,136,148,112,109,114,111)(89,97,121,147,101,123,129,127,120,144,132,139,146,150,99,98,103,125,100,122,134,149,131,142,102,124,145,151,117,126,130,128);(80,81)(82,84)(83,85)(86,87)(88,89,96,103)(90,98,108,128)(91,97,111,125)(92,100,114,121)(93,99,104,130)(94,117,119,146)(95,120,118,102)(101,133,134,112)(105,126,135,150)(106,139,141,151)(107,142,140,127)(109,122,143,147)(110,144,137,124)(113,145,115,132)(116,131,136,129)(123,148,149,138)Polycyclic presentation:
2^8,79|a^2=h,b^2=eh,c^2=d,d^2=fg,e^2=f,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=efg,f^a=fg,g^a=gh,c^b=ch,i^c=i^781528 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 90557);