Finite Group DatabasebetaFinite Group Databasebeta| order | 20096 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | H(5,𝔽2)×C157 ⋊ C4 |
| Sylow | C157 ⋊ #1402 |
122 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 122 conjugacy classes of subgroups — too many to draw. Only its 20 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 173 |
| transitive degree | μt(G) = 2512 |
A minimal faithful permutation representation — 2 generators on 173 points:
(1,3)(2,6)(4,13)(5,12)(7,10)(8,16)(9,15)(11,14)(17,18,21,26,35,48,65,86,111,140,134,107,132,156,150,117,88,63,83,106,131,143,114,89,68,51,38,29,22,27,36,49,66,87,112,141,164,167,160,136,158,153,121,92,67,46,62,81,102,127,147,118,93,72,55,42,31,40,53,70,91,116,145,165,171,173,169,162,155,125,96,71,50,33,45,60,77,98,123,151,122,97,76,59,44,57,74,95,120,149,138,163,170,172,166,157,129,100,75,54,37,24,32,43,56,73,94,119,148,126,101,80,61,78,99,124,152,142,109,137,161,168,159,133,104,79,58,41,28,19,23,30,39,52,69,90,115,144,130,105,82,103,128,154,146,113,84,108,135,139,110,85,64,47,34,25,20);(1,2)(3,10,11,6)(4,8,14,7)(5,9)(12,16,13,15)(18,19,20,22)(21,24,25,31)(23,28,29,27)(26,33,34,44)(30,37,38,40)(32,41,42,36)(35,46,47,61)(39,50,51,57)(43,54,55,53)(45,58,59,49)(48,63,64,82)(52,67,68,78)(56,71,72,74)(60,75,76,70)(62,79,80,66)(65,84,85,107)(69,88,89,103)(73,92,93,99)(77,96,97,95)(81,100,101,91)(83,104,105,87)(86,109,110,136)(90,113,114,132)(94,117,118,128)(98,121,122,124)(102,125,126,120)(106,129,130,116)(108,133,134,112)(111,138,139,162)(115,142,143,158)(119,146,147,156)(123,150,151,154)(127,153,148,152)(131,155,144,149)(135,157,140,145)(137,159,160,141)(161,166,167,165)(163,168,169,164)(170,172,173,171)Polycyclic presentation:
2^7,157|a^2=d,e^2=g,b^a=bc,c^a=ce,e^a=ef,f^a=fg,h^a=h^129,d^b=deg,e^b=eg,f^b=fg,d^c=dfg,e^c=eg,e^d=eg,h^d=h^156680 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20096, 19629);