Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C79 ⋊ #12832 |
358 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 358 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)(84,98)(85,100)(86,99)(87,106)(89,109)(90,97)(91,110)(92,112)(94,116)(95,118)(96,117)(101,130)(102,128)(103,127)(104,125)(105,129)(107,123)(108,134)(111,124)(113,115)(114,136)(119,140)(120,139)(121,141)(122,138)(126,131)(132,142)(133,137)(135,143);(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)(80,82,84,94,85,96,102,120,86,95,103,119,105,121,132,135)(81,87,89,92,91,108,101,114,90,107,111,113,104,133,131,122)(83,93,98,116,100,117,128,139,99,118,127,140,129,141,142,143)(88,106,109,112,110,134,130,136,97,123,124,115,125,137,126,138);(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)(82,92,95,113)(83,97,99,88)(84,101)(85,104)(86,90)(87,94,107,119)(89,102)(91,105)(93,115)(96,122,121,114)(98,126,127,130)(100,110,129,125)(103,131)(106,116)(108,135,133,120)(109,128,124,142)(111,132)(112,118)(117,136)(123,140)(134,143)(137,139)(138,141)Polycyclic presentation:
2^8,79|a^2=f,b^2=d,d^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,i^a=i^78,f^b=fh,i^b=i^78,f^c=fh1300 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 85725);