Finite Group DatabasebetaFinite Group Databasebeta| order | 13428 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C9 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | Dic373⋊C9 |
| derived, Sylow | C373 ⋊ C4×C9 |
| Hall | C373⋊C9 ⋊ C4 |
| Hall | Dic373 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C3 | D373⋊C3 |
| commutator, hypocenter | C373 | C4×C9 |
| Fitting, socle, prime radical, cosocle | C2×C3×C373 | C2×C3 |
| LCS | G ▷ C373 | |
| UCS | C1 ◁ C2×C3 | |
18 conjugacy classes of subgroups.
| minimal degree | μ(G) = 386 |
| transitive degree | μt(G) = 13428 |
A minimal faithful permutation representation — 2 generators on 386 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)(179,274,275,224,225,321)(181,272,273,227,223,266)(184,217,271,229,230,267)(191,285,286,236,237,262)(198,291,292,243,244,257)(203,296,297,249,250,252)(208,300,294,254,255,247)(212,305,289,259,260,240)(215,310,283,263,264,233)(226,298,304,251,277,337)(228,324,325,278,276,318)(231,270,323,280,281,319)(238,330,331,287,288,314)(245,326,334,279,293,309)(256,340,338,301,302,299)(261,342,335,306,307,295)(265,346,332,311,312,290)(268,350,328,315,316,284)(282,322,341,303,327,357)(308,365,362,343,344,339)(313,366,360,347,348,336)(317,367,358,351,352,333)(320,364,345,354,355,329)(349,371,372,359,356,363)(353,373,370,368,369,361)(374,381,380,376,375,382,379,378,377)(383,384,385,386);(1,2,5,11,19,32,51,74,103,138,179,226,249,201,159,122,78,31,48,59,62,64,90,121,157,199,247,295,336,361,368,351,315,230,137,176,190,197,202,207,211,214,216,266,314,325,222,271,310,305,300,296,291,285,272,221,270,320,356,369,372,362,338,297,242,188,132,70,95,117,111,96,69,94,127,165,208,256,303,280,229,182,141,106,77,54,35,22,7,6,12,21,34,53,76,105,140,181,228,279,243,196,154,118,87,55,18,29,39,41,61,86,116,152,193,240,290,333,359,347,311,263,183,102,135,149,155,160,164,167,169,213,262,309,275,323,350,346,342,340,277,326,330,324,274,322,344,348,352,355,357,337,292,235,173,99,130,158,153,146,131,98,129,168,212,261,308,345,328,283,232,185,144,109,80,45,15,26,14,25,40,60,85,115,150,191,238,287,236,189,148,113,83,58,36,10,16,24,38,57,81,110,145,186,233,284,329,343,306,259,210,142,73,100,114,119,123,126,128,166,209,257,304,341,364,367,366,365,327,225,276,288,293,298,302,307,312,316,319,321,334,286,220,134,171,205,200,194,187,172,133,170,215,265,313,349,358,332,289,239,192,151,112,68,28,43,44,27,42,63,89,120,156,198,245,278,227,180,139,104,75,52,33,20,4,8,13,23,37,56,79,108,143,184,231,282,301,254,206,163,107,50,71,84,88,91,93,125,162,204,252,299,339,363,373,371,354,281,178,223,237,244,250,255,260,264,267,269,318,331,273,175,218,258,253,248,241,234,219,174,217,268,317,353,370,360,335,294,246,195,147,97,47,66,82,67,46,65,92,124,161,203,251,224,177,136,101,72,49,30,17,9,3)Polycyclic presentation:
2,3,373,2,3|a^2=d,b^3=e,c^a=c^372,c^b=c^284408 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13428, 2);