Finite Group DatabasebetaFinite Group Databasebeta| order | 13212 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| Fitting, socle, derived | C3×C367 ⋊ C22×C3 |
| Hall | C3×C367⋊C3 ⋊ C22 |
| Sylow | C367 ⋊ C2×C3×D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C367 | C22×C3 |
| LCS | G ▷ C3×C367 | |
| UCS | C1 | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 370 |
| transitive degree | μt(G) = 1101 |
A minimal faithful permutation representation — 2 generators on 370 points:
(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,28,17,18,41)(12,26,22,20,16,13)(19,46,47,30,31,64)(21,44,45,33,29,38)(23,25,43,35,36,39)(32,69,70,49,50,93)(34,67,68,52,48,61)(37,42,66,54,55,62)(40,82,80,58,59,57)(51,98,99,72,73,128)(53,96,97,75,71,90)(56,65,95,77,78,91)(60,111,112,83,84,86)(63,117,109,87,88,81)(74,133,134,101,102,169)(76,131,132,104,100,125)(79,94,130,106,107,126)(85,146,147,113,114,121)(89,153,151,118,119,116)(92,158,144,122,123,110)(103,174,175,136,137,216)(105,172,173,139,135,166)(108,129,171,141,142,167)(115,187,188,148,149,162)(120,194,195,154,155,157)(124,200,192,159,160,152)(127,205,185,163,164,145)(138,221,222,177,178,269)(140,219,220,180,176,213)(143,170,218,182,183,214)(150,234,235,189,190,209)(156,241,242,196,197,204)(161,248,246,201,202,199)(165,253,239,206,207,193)(168,258,232,210,211,186)(179,274,275,224,225,327)(181,272,273,227,223,266)(184,217,271,229,230,267)(191,287,288,236,237,262)(198,293,294,243,244,257)(203,300,301,249,250,252)(208,306,298,254,255,247)(212,311,291,259,260,240)(215,316,285,263,264,233)(226,331,332,277,278,366)(228,330,279,280,276,324)(231,270,329,282,283,325)(238,340,337,289,290,320)(245,343,344,295,296,315)(251,348,349,302,303,310)(256,353,351,307,308,305)(261,356,346,312,313,299)(265,359,341,317,318,292)(268,361,338,321,322,286)(281,367,335,334,333,364)(284,328,345,336,297,365)(304,363,339,326,350,358)(309,362,342,323,354,355)(314,360,347,319,357,352)(368,369,370);(1,2,5,11,19,32,51,74,103,138,179,226,279,222,271,316,311,306,300,293,287,272,221,270,326,302,249,201,159,122,78,31,48,59,62,64,90,121,157,199,247,299,347,351,301,242,188,132,70,95,117,111,96,69,94,127,165,208,256,309,321,230,137,176,190,197,202,207,211,214,216,266,320,336,282,229,182,141,106,77,54,35,22,7,6,12,21,34,53,76,105,140,181,228,281,335,275,329,361,359,356,353,348,343,340,330,274,328,295,243,196,154,118,87,55,18,29,39,41,61,86,116,152,193,240,292,342,349,294,235,173,99,130,158,153,146,131,98,129,168,212,261,314,317,263,183,102,135,149,155,160,164,167,169,213,262,315,358,338,285,232,185,144,109,80,45,15,26,14,25,40,60,85,115,150,191,238,278,333,332,345,350,354,357,360,362,363,365,366,367,331,289,236,189,148,113,83,58,36,10,16,24,38,57,81,110,145,186,233,286,339,344,288,220,134,171,205,200,194,187,172,133,170,215,265,319,312,259,210,142,73,100,114,119,123,126,128,166,209,257,310,355,341,291,239,192,151,112,68,28,43,44,27,42,63,89,120,156,198,245,297,225,276,290,296,303,308,313,318,322,325,327,364,334,280,227,180,139,104,75,52,33,20,4,8,13,23,37,56,79,108,143,184,231,284,337,273,175,218,258,253,248,241,234,219,174,217,268,323,307,254,206,163,107,50,71,84,88,91,93,125,162,204,252,305,352,346,298,246,195,147,97,47,66,82,67,46,65,92,124,161,203,251,304,283,178,223,237,244,250,255,260,264,267,269,324,277,224,177,136,101,72,49,30,17,9,3)(369,370)201 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13212, 27);