Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C79 ⋊ #11771 |
292 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 83 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)(80,83,86,102,87,103,85,101)(81,89,109,135,92,116,91,115)(82,94,98,125,96,124,97,99)(84,100,108,134,106,132,107,133)(88,110,119,138,112,137,111,113)(90,114,120,141,118,140,104,131)(93,121,128,143,122,139,117,129)(95,123,105,136,127,142,126,130);(80,82,87,96)(81,88,92,112)(83,99,103,125)(84,105,106,126)(85,98,86,97)(89,113,116,138)(90,117,118,128)(91,119,109,111)(93,120,122,104)(94,102,124,101)(95,107,127,108)(100,123,132,142)(110,135,137,115)(114,139,140,121)(129,141,143,131)(130,133,136,134);(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)(82,93,96,122)(83,89)(84,104,106,120)(85,91)(86,109)(87,92)(88,95)(90,107,118,108)(94,121,124,139)(97,117,98,128)(99,129,125,143)(100,131,132,141)(101,115)(102,135)(103,116)(105,119)(110,123)(111,126)(112,127)(113,130)(114,133,140,134)(136,138)(137,142)Polycyclic presentation:
2^8,79|a^2=e,b^2=h,c^2=fh,d^2=gh,f^2=h,g^2=h,b^a=bd,d^a=df,g^a=gh,i^a=i^78,c^b=cf,d^b=dg,e^b=efgh,f^b=fh,g^b=gh,e^d=eh1282 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 80115);