Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C79 ⋊ #11869 |
364 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 91 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)(80,83,85,102,87,104,86,103)(81,89,111,134,92,117,91,99)(82,94,97,100,96,127,98,128)(84,107,109,138,108,101,110,137)(88,112,121,142,114,140,113,115)(90,119,122,133,120,116,105,139)(93,123,132,141,124,135,118,125)(95,129,130,143,131,126,106,136);(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)(80,82,87,96)(81,88,92,114)(83,100,104,128)(84,106,108,130)(85,98,86,97)(89,115,117,142)(90,118,120,132)(91,121,111,113)(93,122,124,105)(94,103,127,102)(95,109,131,110)(99,112,134,140)(101,126,107,129)(116,123,119,135)(125,133,141,139)(136,137,143,138);(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)(82,93,96,124)(83,99)(84,105,108,122)(85,91)(86,111)(87,92)(88,95)(89,103)(90,109,120,110)(94,125,127,141)(97,118,98,132)(100,135,128,123)(101,119,107,116)(102,117)(104,134)(106,121)(112,136)(113,130)(114,131)(115,129)(126,142)(133,137,139,138)(140,143)Polycyclic presentation:
2^8,79|a^2=e,b^2=h,c^2=gh,d^2=gh,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,i^b=i^78,d^c=dh,e^d=eh1750 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 80567);