Finite Group DatabasebetaFinite Group Databasebeta| order | 31314 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C17⋊(D307⋊C3) |
| unambiguous semidirect product | (C307⋊C3)⋊D17 |
| unambiguous semidirect product | D5219⋊C3 |
| Fitting, socle, derived, Hall | C17×C307 ⋊ C2×C3 |
| Hall | C17×C307⋊C3 ⋊ C2 |
| Hall | D5219 ⋊ C3 |
| Hall | C307⋊C3 ⋊ D17 |
| Sylow | C307 ⋊ C3×D17 |
| Sylow | C17 ⋊ D307⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C17×C307 | C2×C3 |
| LCS | G ▷ C17×C307 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 324 |
| transitive degree | μt(G) = 5219 |
A minimal faithful permutation representation — 2 generators on 324 points:
(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(19,23,24,20,21,25)(22,31,32,26,27,41)(28,44,45,34,35,58)(29,43,39,37,33,30)(36,63,64,47,48,81)(38,61,62,50,46,55)(40,42,60,52,53,56)(49,86,87,66,67,110)(51,84,85,69,65,78)(54,59,83,71,72,79)(57,99,97,75,76,74)(68,115,116,89,90,145)(70,113,114,92,88,107)(73,82,112,94,95,108)(77,128,129,100,101,103)(80,134,126,104,105,98)(91,150,151,118,119,186)(93,148,149,121,117,142)(96,111,147,123,124,143)(102,163,164,130,131,138)(106,170,168,135,136,133)(109,175,161,139,140,127)(120,191,192,153,154,231)(122,189,190,156,152,183)(125,146,188,158,159,184)(132,202,203,165,166,179)(137,209,210,171,172,174)(141,215,207,176,177,169)(144,220,200,180,181,162)(155,235,236,194,195,275)(157,233,234,197,193,228)(160,187,232,196,199,229)(167,242,243,204,205,224)(173,249,250,211,212,219)(178,256,254,216,217,214)(182,261,247,221,222,208)(185,266,240,225,226,201)(198,276,277,238,237,274)(206,278,279,244,245,270)(213,282,283,251,252,265)(218,289,290,257,258,260)(223,295,287,262,263,255)(227,300,281,267,268,248)(230,305,239,271,272,241)(246,306,307,280,273,308)(253,309,310,284,285,304)(259,311,312,291,292,299)(264,315,314,296,297,294)(269,320,286,301,302,288)(293,321,322,313,303,323)(298,324,318,316,317,319);(1,2,4,6,8,10,12,14,16,17,15,13,11,9,7,5,3)(18,19,22,28,36,49,68,91,120,155,196,158,123,94,71,52,39,24,23,29,38,51,70,93,122,157,198,239,240,200,161,126,97,62,32,43,31,42,57,77,102,132,167,206,246,281,247,207,168,129,85,45,60,61,44,59,80,106,137,173,213,253,286,287,254,210,164,114,64,83,99,84,63,82,109,141,178,218,259,293,314,290,250,203,149,87,112,134,128,113,86,111,144,182,223,264,298,318,312,283,243,190,116,147,175,170,163,148,115,146,185,227,269,303,317,322,310,279,234,151,188,220,215,209,202,189,150,187,230,273,285,292,297,302,307,277,192,232,266,261,256,249,242,233,191,195,237,245,252,258,263,268,272,236,275,305,300,295,289,282,278,276,235,154,193,205,212,217,222,226,229,231,274,308,320,315,311,309,306,271,199,119,152,166,172,177,181,184,186,228,270,304,323,324,321,301,267,225,159,90,117,131,136,140,143,145,183,224,265,299,319,316,296,262,221,180,124,67,88,101,105,108,110,142,179,219,260,294,313,291,257,216,176,139,95,48,65,76,79,81,107,138,174,214,255,288,284,251,211,171,135,104,72,35,46,56,58,78,103,133,169,208,248,280,244,204,165,130,100,75,53,27,33,41,55,74,98,127,162,201,241,238,197,156,121,92,69,50,37,21,25,30,40,54,73,96,125,160,194,153,118,89,66,47,34,26,20)Polycyclic presentation:
2,3,17,307|c^a=c^16,d^a=d^306,d^b=d^289897 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31314, 17);