Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C53 ⋊ SD256 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #20272927 |
| commutator, cosocle | C53×C64 | C22 |
| Frattini | C64 | C2×D53 |
| Fitting, prime radical | C53×D64 | C2 |
| socle | C2×C53 | D64 |
| hypercenter | D64 | D53 |
| hypocenter | C53 | SD256 |
| LCS | G ▷ C53×C64 ▷ C32×C53 ▷ C16×C53 ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ C64 ◁ D64 | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 181 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 2 generators on 181 points:
(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(55,62)(56,69)(57,70)(58,76)(59,81)(60,85)(63,64)(65,103)(66,107)(67,109)(68,92)(71,117)(72,121)(73,124)(74,93)(75,88)(77,132)(78,136)(79,98)(80,89)(82,141)(83,102)(84,90)(86,106)(87,91)(94,126)(95,125)(96,145)(97,101)(99,142)(100,144)(104,168)(105,167)(108,169)(110,111)(112,118)(113,122)(114,115)(116,133)(119,176)(120,156)(123,159)(127,170)(128,149)(129,130)(131,147)(134,163)(135,148)(137,166)(138,165)(139,152)(140,151)(143,154)(146,150)(153,155)(157,180)(158,179)(160,178)(161,162)(164,181)(171,175)(172,174)(173,177);(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)(54,55,61,68)(56,62,74,92)(57,75,79,128)(58,80,83,139)(59,84,86,143)(60,87,85,67)(63,93,97,69)(64,98,101,70)(65,102,105,76)(66,106,108,81)(71,103,120,167)(72,107,123,169)(73,109,122,91)(77,131,134,180)(78,135,133,127)(82,140,129,138)(88,146,149,111)(89,150,152,110)(90,153,154,118)(94,156,158,117)(95,159,160,121)(96,113,144,124)(99,163,164,132)(100,116,145,136)(104,130,137,141)(112,142,155,181)(114,126,161,179)(115,125,162,178)(119,168,171,166)(147,176,157,175)(148,173,170,174)(151,172,165,177)Polycyclic presentation:
2^8,53|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^52,c^b=cd,d^b=de,e^b=ef,f^b=fg,g^b=gh977 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 57483);