Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C79 ⋊ #12597 |
258 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 127 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)(82,93,94,118)(84,98)(85,100)(86,99)(87,105,92,115)(89,109)(90,111)(91,110)(95,119,104,129)(96,97,120,123)(101,126)(102,125)(103,127)(106,132,116,139)(107,108,117,122)(112,137)(113,136)(114,131)(121,124,130,134)(128,142)(133,135,140,141)(138,143);(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)(81,87,90,92)(83,97,99,123)(84,95)(85,96)(86,94)(88,108)(89,106,112,116)(91,107,114,117)(93,100,118,127)(98,124,125,134)(101,121)(102,104)(103,120)(105,131)(109,135)(110,115)(111,122)(113,133,138,140)(119,126,129,142)(128,130)(132,143)(136,139)(137,141);(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)(80,81,84,89,85,91,101,113,86,90,102,112,103,114,128,138)(82,92,95,116,96,117,121,140,94,87,104,106,120,107,130,133)(83,88,98,109,100,110,126,136,99,111,125,137,127,131,142,143)(93,115,119,139,97,122,124,141,118,105,129,132,123,108,134,135)Polycyclic presentation:
2^8,79|a^2=e,b^2=f,c^2=d,e^2=g,g^2=h,b^a=bd,f^a=fh,i^a=i^78,c^b=cg,d^b=dh,i^b=i^78,f^c=fh1780 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 84593);