Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C79 ⋊ #12915 |
374 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 374 conjugacy classes of subgroups — too many to draw. Only its 147 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)(81,88)(82,92)(84,97)(85,98)(86,96)(87,102)(89,107)(90,108)(91,106)(93,113)(94,114)(95,115)(99,121)(100,119)(101,120)(103,112)(104,124)(105,125)(109,131)(110,129)(111,130)(116,123)(117,135)(118,136)(122,138)(126,133)(127,134)(128,140)(132,142)(137,139)(141,143);(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)(80,82,84,94,85,95,99,118,86,93,100,116,101,117,122,137)(81,87,89,104,90,105,109,128,91,103,110,126,111,127,132,141)(83,92,97,114,98,115,121,136,96,113,119,123,120,135,138,139)(88,102,107,124,108,125,131,140,106,112,129,133,130,134,142,143);(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)(80,81,84,89,85,90,99,109,86,91,100,110,101,111,122,132)(82,87,94,104,95,105,118,128,93,103,116,126,117,127,137,141)(83,88,97,107,98,108,121,131,96,106,119,129,120,130,138,142)(92,112,114,133,115,134,136,143,113,102,123,124,135,125,139,140)Polycyclic presentation:
2^8,79|a^2=f,b^2=f,f^2=g,g^2=h,b^a=bd,c^a=ce,i^a=i^78,e^b=eh,i^b=i^78,d^c=dh1780 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 86045);