Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C79 ⋊ #11871 |
360 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 99 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)(80,83,86,103,87,104,85,102)(81,89,91,99,92,118,111,134)(82,94,97,127,98,128,96,100)(84,107,108,138,110,101,109,137)(88,112,114,116,115,141,113,140)(90,120,105,139,122,117,121,133)(93,123,119,125,124,135,132,142)(95,129,106,136,131,126,130,143);(80,82,87,98)(81,88,92,115)(83,100,104,127)(84,106,110,130)(85,97,86,96)(89,116,118,140)(90,119,122,132)(91,113,111,114)(93,121,124,105)(94,103,128,102)(95,109,131,108)(99,112,134,141)(101,126,107,129)(117,123,120,135)(125,133,142,139)(136,138,143,137);(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)(80,81,85,91,87,92,86,111)(82,93,96,119,98,124,97,132)(83,99,102,118,104,134,103,89)(84,105,108,90,110,121,109,122)(88,106,113,131,115,130,114,95)(94,125,100,135,128,142,127,123)(101,120,137,133,107,117,138,139)(112,126,140,143,141,129,116,136)Polycyclic presentation:
2^8,79|a^2=e,b^2=h,c^2=g,d^2=gh,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=cg,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,d^c=dh,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, 80574);