Finite Group DatabasebetaFinite Group Databasebeta| order | 26850 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C5 |
| unambiguous semidirect product | C179⋊((C5⋊D5)⋊C3) |
| unambiguous semidirect product | (C5⋊D895)⋊C3 |
| Fitting, socle, derived, Hall | C52×C179 ⋊ C2×C3 |
| Hall | C52⋊C3×C179 ⋊ C2 |
| Hall | C5⋊D895 ⋊ C3 |
| Sylow | C179 ⋊ (C5⋊D5)⋊C3 |
| Hall | C52⋊C3 ⋊ D179 |
| Sylow | C52 ⋊ C3×D179 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C52×C179 | C2×C3 |
| LCS | G ▷ C52×C179 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 194 |
| transitive degree | μt(G) = 2685 |
A minimal faithful permutation representation — 2 generators on 194 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)(54,55)(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)(180,181,183)(182,192,188,185,186,194)(184,189,193,190,191,187);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,100,102,104,106,108,110,112,114,116,118,120,122,124,126,128,130,132,134,136,138,140,142,144,146,148,150,152,154,156,158,160,162,164,166,168,170,172,174,176,178,179,177,175,173,171,169,167,165,163,161,159,157,155,153,151,149,147,145,143,141,139,137,135,133,131,129,127,125,123,121,119,117,115,113,111,109,107,105,103,101,99,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(180,182,187,193,185)(181,184,186,192,190)(183,188,189,191,194)Polycyclic presentation:
2,3,5^2,179|c^a=c^4,d^a=d^4,e^a=e^178,c^b=cd,d^b=c^2d^3989 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26850, 14);