Finite Group DatabasebetaFinite Group Databasebeta| order | 34998 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle, derived, Hall | C19×C307 ⋊ C2×C3 |
| Hall | #31775847 ⋊ C2 |
| Hall | D5833 ⋊ C3 |
| Sylow | C307 ⋊ D19⋊C3 |
| Sylow | C19 ⋊ D307⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C19×C307 | C2×C3 |
| LCS | G ▷ C19×C307 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 326 |
| transitive degree | μt(G) = 5833 |
A minimal faithful permutation representation — 2 generators on 326 points:
(2,6,7,3,4,8)(5,14,11,9,10,16)(12,17,19,18,15,13)(21,25,26,22,23,27)(24,33,34,28,29,43)(30,46,47,36,37,60)(31,45,41,39,35,32)(38,65,66,49,50,83)(40,63,64,52,48,57)(42,44,62,54,55,58)(51,88,89,68,69,112)(53,86,87,71,67,80)(56,61,85,73,74,81)(59,101,99,77,78,76)(70,117,118,91,92,147)(72,115,116,94,90,109)(75,84,114,96,97,110)(79,130,131,102,103,105)(82,136,128,106,107,100)(93,152,153,120,121,188)(95,150,151,123,119,144)(98,113,149,125,126,145)(104,165,166,132,133,140)(108,172,170,137,138,135)(111,177,163,141,142,129)(122,193,194,155,156,233)(124,191,192,158,154,185)(127,148,190,160,161,186)(134,204,205,167,168,181)(139,211,212,173,174,176)(143,217,209,178,179,171)(146,222,202,182,183,164)(157,237,238,196,197,277)(159,235,236,199,195,230)(162,189,234,198,201,231)(169,244,245,206,207,226)(175,251,252,213,214,221)(180,258,256,218,219,216)(184,263,249,223,224,210)(187,268,242,227,228,203)(200,278,279,240,239,276)(208,280,281,246,247,272)(215,284,285,253,254,267)(220,291,292,259,260,262)(225,297,289,264,265,257)(229,302,283,269,270,250)(232,307,241,273,274,243)(248,308,309,282,275,310)(255,311,312,286,287,306)(261,313,314,293,294,301)(266,317,316,298,299,296)(271,322,288,303,304,290)(295,323,324,315,305,325)(300,326,320,318,319,321);(1,2,5,11,17,14,18,4,8,13,19,7,6,12,10,15,16,9,3)(20,21,24,30,38,51,70,93,122,157,198,160,125,96,73,54,41,26,25,31,40,53,72,95,124,159,200,241,242,202,163,128,99,64,34,45,33,44,59,79,104,134,169,208,248,283,249,209,170,131,87,47,62,63,46,61,82,108,139,175,215,255,288,289,256,212,166,116,66,85,101,86,65,84,111,143,180,220,261,295,316,292,252,205,151,89,114,136,130,115,88,113,146,184,225,266,300,320,314,285,245,192,118,149,177,172,165,150,117,148,187,229,271,305,319,324,312,281,236,153,190,222,217,211,204,191,152,189,232,275,287,294,299,304,309,279,194,234,268,263,258,251,244,235,193,197,239,247,254,260,265,270,274,238,277,307,302,297,291,284,280,278,237,156,195,207,214,219,224,228,231,233,276,310,322,317,313,311,308,273,201,121,154,168,174,179,183,186,188,230,272,306,325,326,323,303,269,227,161,92,119,133,138,142,145,147,185,226,267,301,321,318,298,264,223,182,126,69,90,103,107,110,112,144,181,221,262,296,315,293,259,218,178,141,97,50,67,78,81,83,109,140,176,216,257,290,286,253,213,173,137,106,74,37,48,58,60,80,105,135,171,210,250,282,246,206,167,132,102,77,55,29,35,43,57,76,100,129,164,203,243,240,199,158,123,94,71,52,39,23,27,32,42,56,75,98,127,162,196,155,120,91,68,49,36,28,22)Polycyclic presentation:
2,3,19,307|c^a=c^18,d^a=d^306,c^b=c^7,d^b=d^289978 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(34998, 23);