Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C79×D64 ⋊ C2 |
| Sylow | C79 ⋊ #13741 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D32×D79 |
| commutator, cosocle | C32×C79 | C23 |
| Frattini | C32 | C22×D79 |
| Fitting | C79×D64 | C2 |
| socle | C2×C79 | C2×D32 |
| hypercenter | D64 | D79 |
| hypocenter | C79 | #13741 |
| LCS | G ▷ C32×C79 ▷ C16×C79 ▷ C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ D64 | |
128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 207 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 3 generators on 207 points:
(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)(80,83,88,87)(81,90,94,119)(82,96,100,120)(84,101,110,109)(85,102,115,114)(86,103,118,113)(89,121,125,124)(91,126,130,179)(92,127,132,171)(93,116,134,177)(95,135,138,137)(97,139,142,180)(98,140,144,173)(99,117,146,178)(104,147,156,169)(105,148,159,170)(106,149,162,161)(107,150,165,164)(108,151,168,163)(111,152,172,175)(112,153,174,176)(122,181,184,183)(123,182,185,133)(128,186,189,188)(129,154,192,203)(131,187,194,193)(136,195,196,145)(141,157,198,204)(143,197,199,191)(155,166,201,206)(158,167,202,207)(160,200,205,190);(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)(80,82)(81,89)(83,96)(84,105)(85,112)(86,117)(87,120)(88,100)(90,121)(91,128)(92,131)(93,133)(94,125)(95,104)(97,107)(98,106)(99,113)(101,148)(102,153)(103,146)(108,167)(109,170)(110,159)(111,122)(114,176)(115,174)(116,123)(118,178)(119,124)(126,186)(127,187)(129,191)(130,189)(132,194)(134,182)(135,147)(136,166)(137,169)(138,156)(139,150)(140,149)(141,190)(142,165)(143,154)(144,162)(145,155)(151,202)(152,181)(157,160)(158,163)(161,173)(164,180)(168,207)(171,193)(172,184)(175,183)(177,185)(179,188)(192,197)(195,201)(196,206)(198,200)(199,203)(204,205);(80,81)(82,95)(83,90)(84,104)(85,111)(86,116)(87,119)(88,94)(89,97)(91,107)(92,106)(93,113)(96,135)(98,143)(99,145)(100,138)(101,147)(102,152)(103,134)(105,131)(108,166)(109,169)(110,156)(112,128)(114,175)(115,172)(117,136)(118,177)(120,137)(121,139)(122,158)(123,157)(124,180)(125,142)(126,150)(127,149)(129,190)(130,165)(132,162)(133,141)(140,197)(144,199)(146,195)(148,187)(151,201)(153,186)(154,160)(155,163)(159,194)(161,171)(164,179)(167,181)(168,206)(170,193)(173,191)(174,189)(176,188)(178,196)(182,198)(183,207)(184,202)(185,204)(192,200)(203,205)Polycyclic presentation:
2^8,79|c^2=h,d^2=ef,e^2=fg,f^2=gh,g^2=h,b^a=bd,d^a=de,e^a=ef,f^a=fg,g^a=gh,d^b=de,e^b=ef,f^b=fg,g^b=gh,i^c=i^781435 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 90619);