Finite Group DatabasebetaFinite Group Databasebeta| order | 25216 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D12608 |
| Fitting | C64×C197 ⋊ C2 |
| Sylow | C197 ⋊ D64 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D6304 |
| commutator, cosocle | C32×C197 | C22 |
| Frattini | C32 | C2×D197 |
| Fitting | C64×C197 | C2 |
| socle | C2×C197 | D32 |
| hypercenter | C64 | D197 |
| hypocenter | C197 | D64 |
| LCS | G ▷ C32×C197 ▷ C16×C197 ▷ C8×C197 ▷ C4×C197 ▷ C2×C197 ▷ C197 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ C64 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 261 |
| transitive degree | μt(G) = 12608 |
A minimal faithful permutation representation — 2 generators on 261 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)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261);(2,64)(3,63)(4,62)(5,61)(6,60)(7,59)(8,58)(9,57)(10,56)(11,55)(12,54)(13,53)(14,52)(15,51)(16,50)(17,49)(18,48)(19,47)(20,46)(21,45)(22,44)(23,43)(24,42)(25,41)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(66,261)(67,260)(68,259)(69,258)(70,257)(71,256)(72,255)(73,254)(74,253)(75,252)(76,251)(77,250)(78,249)(79,248)(80,247)(81,246)(82,245)(83,244)(84,243)(85,242)(86,241)(87,240)(88,239)(89,238)(90,237)(91,236)(92,235)(93,234)(94,233)(95,232)(96,231)(97,230)(98,229)(99,228)(100,227)(101,226)(102,225)(103,224)(104,223)(105,222)(106,221)(107,220)(108,219)(109,218)(110,217)(111,216)(112,215)(113,214)(114,213)(115,212)(116,211)(117,210)(118,209)(119,208)(120,207)(121,206)(122,205)(123,204)(124,203)(125,202)(126,201)(127,200)(128,199)(129,198)(130,197)(131,196)(132,195)(133,194)(134,193)(135,192)(136,191)(137,190)(138,189)(139,188)(140,187)(141,186)(142,185)(143,184)(144,183)(145,182)(146,181)(147,180)(148,179)(149,178)(150,177)(151,176)(152,175)(153,174)(154,173)(155,172)(156,171)(157,170)(158,169)(159,168)(160,167)(161,166)(162,165)(163,164)Polycyclic presentation:
2^7,197|c^2=de,d^2=ef,e^2=fg,f^2=g,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,h^a=h^196,c^b=cd,d^b=de,e^b=ef,f^b=fg,h^b=h^1966307 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(25216, 2723);