Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C79×C128 ⋊ C2 |
| Sylow | C79 ⋊ SD256 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D5056 |
| commutator, cosocle | C64×C79 | C22 |
| Frattini | C64 | C2×D79 |
| Fitting | C79×C128 | C2 |
| socle | C2×C79 | D64 |
| prime radical | D5056 | C2 |
| hypercenter | C128 | D79 |
| hypocenter | C79 | SD256 |
| LCS | G ▷ C64×C79 ▷ C32×C79 ▷ C16×C79 ▷ C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ C64 ◁ C128 | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 207 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 2 generators on 207 points:
(1,2)(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)(81,88)(82,95)(83,96)(84,102)(85,107)(86,111)(89,90)(91,129)(92,133)(93,135)(94,118)(97,143)(98,147)(99,150)(100,119)(101,114)(103,158)(104,162)(105,124)(106,115)(108,167)(109,128)(110,116)(112,132)(113,117)(120,152)(121,151)(122,171)(123,127)(125,168)(126,170)(130,194)(131,193)(134,195)(136,137)(138,144)(139,148)(140,141)(142,159)(145,202)(146,182)(149,185)(153,196)(154,175)(155,156)(157,173)(160,189)(161,174)(163,192)(164,191)(165,178)(166,177)(169,180)(172,176)(179,181)(183,206)(184,205)(186,204)(187,188)(190,207)(197,201)(198,200)(199,203);(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)(80,81,87,94)(82,88,100,118)(83,101,105,154)(84,106,109,165)(85,110,112,169)(86,113,111,93)(89,119,123,95)(90,124,127,96)(91,128,131,102)(92,132,134,107)(97,129,146,193)(98,133,149,195)(99,135,148,117)(103,157,160,206)(104,161,159,153)(108,166,155,164)(114,172,175,137)(115,176,178,136)(116,179,180,144)(120,182,184,143)(121,185,186,147)(122,139,170,150)(125,189,190,158)(126,142,171,162)(130,156,163,167)(138,168,181,207)(140,152,187,205)(141,151,188,204)(145,194,197,192)(173,202,183,201)(174,199,196,200)(177,198,191,203)Polycyclic presentation:
2^8,79|a^2=h,c^2=de,d^2=ef,e^2=fg,f^2=gh,g^2=h,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,g^a=gh,i^a=i^78,c^b=cd,d^b=de,e^b=ef,f^b=fg,g^b=gh,i^b=i^785059 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 57484);