Finite Group DatabasebetaFinite Group Databasebeta| order | 33762 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C17⋊(D331⋊C3) |
| unambiguous semidirect product | (C331⋊C3)⋊D17 |
| unambiguous semidirect product | D5627⋊C3 |
| Fitting, socle, derived, Hall | C17×C331 ⋊ C2×C3 |
| Hall | C17×C331⋊C3 ⋊ C2 |
| Hall | D5627 ⋊ C3 |
| Hall | C331⋊C3 ⋊ D17 |
| Sylow | C331 ⋊ C3×D17 |
| Sylow | C17 ⋊ D331⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C17×C331 | C2×C3 |
| LCS | G ▷ C17×C331 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 348 |
| transitive degree | μt(G) = 5627 |
A minimal faithful permutation representation — 2 generators on 348 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)(120,191,192,153,154,233)(122,189,190,156,152,183)(125,146,188,158,159,184)(132,204,205,165,166,179)(137,211,212,171,172,174)(141,217,209,176,177,169)(144,222,202,180,181,162)(155,238,239,194,195,285)(157,236,237,197,193,230)(160,187,235,199,200,231)(167,250,251,206,207,226)(173,257,258,213,214,221)(178,264,262,218,219,216)(182,269,255,223,224,210)(185,274,248,227,228,203)(196,290,291,241,242,331)(198,288,289,243,240,282)(201,234,287,245,246,283)(208,298,299,252,253,278)(215,305,306,259,260,273)(220,312,313,265,266,268)(225,318,310,270,271,263)(229,322,303,275,276,256)(232,325,296,279,280,249)(244,333,334,293,292,329)(247,286,332,295,294,330)(254,336,337,300,301,327)(261,340,341,307,308,324)(267,344,345,314,315,321)(272,348,347,319,320,317)(277,346,343,316,323,311)(281,342,339,309,326,304)(284,338,335,302,328,297);(1,2,4,6,8,10,12,14,16,17,15,13,11,9,7,5,3)(18,19,22,28,36,49,68,91,120,155,196,242,292,301,308,315,320,323,326,328,330,331,295,245,199,158,123,94,71,52,39,24,23,29,38,51,70,93,122,157,198,244,294,195,240,253,260,266,271,276,280,283,285,329,335,296,248,202,161,126,97,62,32,43,31,42,57,77,102,132,167,208,254,302,246,154,193,207,214,219,224,228,231,233,282,327,339,303,255,209,168,129,85,45,60,61,44,59,80,106,137,173,215,261,309,279,200,119,152,166,172,177,181,184,186,230,278,324,343,310,262,212,164,114,64,83,99,84,63,82,109,141,178,220,267,316,275,227,159,90,117,131,136,140,143,145,183,226,273,321,347,313,258,205,149,87,112,134,128,113,86,111,144,182,225,272,319,270,223,180,124,67,88,101,105,108,110,142,179,221,268,317,345,306,251,190,116,147,175,170,163,148,115,146,185,229,277,314,265,218,176,139,95,48,65,76,79,81,107,138,174,216,263,311,341,299,237,151,188,222,217,211,204,189,150,187,232,281,307,259,213,171,135,104,72,35,46,56,58,78,103,133,169,210,256,304,337,289,192,235,274,269,264,257,250,236,191,234,284,300,252,206,165,130,100,75,53,27,33,41,55,74,98,127,162,203,249,297,334,239,287,325,322,318,312,305,298,288,238,286,293,243,197,156,121,92,69,50,37,21,25,30,40,54,73,96,125,160,201,247,291,332,338,342,346,348,344,340,336,333,290,241,194,153,118,89,66,47,34,26,20)Polycyclic presentation:
2,3,17,331|c^a=c^16,d^a=d^330,d^b=d^299965 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33762, 12);