Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C79 ⋊ #12877 |
366 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 366 conjugacy classes of subgroups — too many to draw. Only its 119 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:
(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)(80,83,85,108,86,98,104,100)(81,88,107,112,89,113,91,117)(82,93,96,128,95,121,126,123)(84,103,105,99,106,137,138,130)(87,109,119,140,92,118,111,142)(90,116,133,129,114,132,101,97)(94,125,127,122,102,135,136,131)(110,141,124,120,115,143,134,139);(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)(81,87)(83,93)(84,102)(85,96)(86,95)(88,109)(89,92)(90,115)(91,111)(94,106)(97,120)(98,121)(99,131)(100,123)(101,124)(103,135)(104,126)(105,136)(107,119)(108,128)(110,114)(112,140)(113,118)(116,143)(117,142)(122,130)(125,137)(127,138)(129,139)(132,141)(133,134);(80,81)(82,92,95,87)(83,97)(84,101)(85,107)(86,89)(88,99)(90,105)(91,104)(93,120,121,139)(94,124,102,134)(96,111,126,119)(98,129)(100,132)(103,117)(106,133)(108,116)(109,122,118,131)(110,127,115,136)(112,137)(113,130)(114,138)(123,141,128,143)(125,140,135,142)Polycyclic presentation:
2^8,79|a^2=f,c^2=dg,e^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,g^a=gh,e^b=eh,f^b=fh,d^c=dh,e^c=eg,g^c=gh,i^c=i^781294 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 85903);