Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C53 ⋊ Q256 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #20272927 |
| commutator, cosocle | C53×C64 | C22 |
| Frattini | C64 | C2×D53 |
| Fitting | C53×Q128 | C2 |
| socle, prime radical | C2×C53 | D64 |
| hypercenter | Q128 | D53 |
| hypocenter | C53 | Q256 |
| LCS | G ▷ C53×C64 ▷ C32×C53 ▷ C16×C53 ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ C64 ◁ Q128 | |
42 conjugacy classes of subgroups.
| minimal degree | μ(G) = 309 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 2 generators on 309 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)(54,56,62,76)(55,63,69,107)(57,78,84,153)(58,86,90,179)(59,92,95,193)(60,97,99,199)(61,101,98,75)(64,109,113,221)(65,115,118,236)(66,120,122,244)(67,124,125,247)(68,126,100,106)(70,127,131,77)(71,108,102,80)(72,114,138,79)(73,119,141,87)(74,123,143,93)(81,162,165,285)(82,167,169,288)(83,171,168,152)(85,103,175,209)(88,183,185,296)(89,187,184,178)(91,104,190,212)(94,195,180,192)(96,105,197,214)(110,223,225,303)(111,227,228,306)(112,229,200,220)(116,238,239,309)(117,240,201,235)(121,245,188,243)(128,252,254,161)(129,255,256,166)(130,148,144,170)(132,222,204,276)(133,226,205,275)(134,160,202,218)(135,249,259,145)(136,237,261,278)(137,157,203,232)(139,250,265,146)(140,181,191,241)(142,251,267,147)(149,216,210,154)(150,215,206,158)(151,246,274,163)(155,177,230,294)(156,176,231,293)(159,198,217,213)(164,287,279,284)(172,234,291,308)(173,233,292,307)(174,248,186,208)(182,207,295,298)(189,242,194,211)(196,219,297,301)(224,305,268,260)(253,282,263,286)(257,304,271,302)(258,269,262,289)(264,270,266,281)(272,277,290,300)(273,280,283,299);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,55,62,69)(56,70,76,131)(57,77,84,127)(58,85,90,175)(59,91,95,190)(60,96,99,197)(61,100,98,68)(63,102,107,71)(64,108,113,80)(65,114,118,79)(66,119,122,87)(67,123,125,93)(72,135,138,259)(73,139,141,265)(74,142,143,267)(75,144,101,130)(78,115,153,236)(81,161,165,252)(82,166,169,255)(83,170,168,148)(86,109,179,221)(88,182,185,295)(89,186,184,174)(92,128,193,254)(94,194,180,189)(97,129,199,256)(103,206,209,150)(104,210,212,149)(105,213,214,198)(106,134,126,202)(110,222,225,276)(111,226,228,275)(112,160,200,218)(116,237,239,278)(117,157,201,232)(120,204,244,132)(121,181,188,241)(124,205,247,133)(136,260,261,305)(137,262,203,258)(140,266,191,264)(145,216,249,154)(146,215,250,158)(147,246,251,163)(151,238,274,309)(152,240,171,235)(155,173,230,292)(156,172,231,291)(159,196,217,297)(162,233,285,307)(164,286,279,282)(167,234,288,308)(176,223,293,303)(177,227,294,306)(178,229,187,220)(183,219,296,301)(192,263,195,253)(207,287,298,284)(208,273,248,283)(211,272,242,290)(224,304,268,302)(243,257,245,271)(269,277,289,300)(270,280,281,299)Polycyclic presentation:
2^8,53|a^2=h,b^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,c^b=cd,d^b=de,e^b=ef,f^b=fg,g^b=gh,i^b=i^52977 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 57485);