Finite Group DatabasebetaFinite Group Databasebeta| order | 12892 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C11⋊#11924078 |
| Fitting, socle, derived, Hall | C11×C293 ⋊ C4 |
| Sylow | C293 ⋊ Dic11 |
| Sylow | C11 ⋊ #11924078 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C11×C293 | C4 |
| prime radical, cosocle | C11×D293 | C2 |
| LCS | G ▷ C11×C293 | |
| UCS | C1 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 304 |
| transitive degree | μt(G) = 3223 |
A minimal faithful permutation representation — 2 generators on 304 points:
(2,3,4,6)(5,8,9,15)(7,12,13,11)(10,17,18,28)(14,21,22,24)(16,25,26,20)(19,30,31,45)(23,34,35,41)(27,38,39,37)(29,42,43,33)(32,47,48,66)(36,51,52,62)(40,55,56,58)(44,59,60,54)(46,63,64,50)(49,68,69,91)(53,72,73,87)(57,76,77,83)(61,80,81,79)(65,84,85,75)(67,88,89,71)(70,93,94,120)(74,97,98,116)(78,101,102,112)(82,105,106,108)(86,109,110,104)(90,113,114,100)(92,117,118,96)(95,122,123,153)(99,126,127,149)(103,130,131,145)(107,134,135,141)(111,138,139,137)(115,142,143,133)(119,146,147,129)(121,150,151,125)(124,155,156,187)(128,159,160,183)(132,162,157,179)(136,164,165,175)(140,168,169,171)(144,172,173,167)(148,176,177,163)(152,180,181,161)(154,184,185,158)(166,192,193,207)(170,196,197,203)(174,200,201,199)(178,204,205,195)(182,208,209,191)(186,211,212,190)(188,194,214,189)(198,219,220,230)(202,223,224,226)(206,227,228,222)(210,231,232,218)(213,234,235,217)(215,221,237,216)(225,242,243,249)(229,246,247,245)(233,250,251,241)(236,253,254,240)(238,244,256,239)(248,261,262,264)(252,265,266,260)(255,268,269,259)(257,263,271,258)(267,276,277,275)(270,279,280,274)(272,278,282,273)(281,286,287,285)(283,288,289,284)(290,292,293,291)(295,296)(297,298)(299,300)(301,302)(303,304);(1,2,5,10,19,32,49,70,95,124,157,131,102,77,56,39,26,15,24,37,54,75,100,129,161,190,216,220,197,169,139,110,85,64,45,62,83,108,137,167,195,218,240,258,262,247,228,205,177,147,118,91,116,145,175,203,226,245,260,274,284,287,280,269,254,235,212,185,153,183,155,188,215,238,257,272,283,290,292,286,276,261,242,219,192,162,126,93,121,152,182,210,233,252,267,278,268,250,227,200,168,134,101,72,47,67,90,115,144,174,202,225,244,234,208,176,142,109,80,55,34,17,29,44,61,82,107,136,166,194,184,150,117,88,63,42,25,12,3,7,14,23,36,53,74,99,128,160,127,98,73,52,35,22,13,6,11,20,33,50,71,96,125,158,189,193,165,135,106,81,60,43,28,41,58,79,104,133,163,191,217,239,243,224,201,173,143,114,89,66,87,112,141,171,199,222,241,259,273,277,266,251,232,209,181,151,120,149,179,207,230,249,264,275,285,291,293,289,282,271,256,237,214,187,159,122,154,186,213,236,255,270,281,288,279,265,246,223,196,164,130,97,68,92,119,148,178,206,229,248,263,253,231,204,172,138,105,76,51,30,46,65,86,111,140,170,198,221,211,180,146,113,84,59,38,21,8,16,27,40,57,78,103,132,156,123,94,69,48,31,18,9,4)(294,295,297,299,301,303,304,302,300,298,296)Polycyclic presentation:
2^2,11,293|a^2=b,c^a=c^10,d^a=d^138,d^b=d^292817 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12892, 7);