Finite Group DatabasebetaFinite Group Databasebeta| order | 31213 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C73×C13 · Schur multiplier (H2) = C72 |
| direct product | C13×#14558883 |
| subgroup | G/H | |
|---|---|---|
| center | C13×C49 | C72 |
| commutator, Frattini, cosocle | C7 | C73×C13 |
| socle | C7×C13 | C73 |
| prime radical | C13×H(3,𝔽7) | C7 |
| LCS | G ▷ C7 ▷ C1 | |
| UCS | C1 ◁ C13×C49 ◁ G | |
250 conjugacy classes of subgroups.
| minimal degree | μ(G) = 356 |
| transitive degree | μt(G) = 4459 |
A minimal faithful permutation representation — 3 generators on 356 points:
(1,2,5,12,26,51,85)(3,7,14,28,53,87,129)(4,8,15,29,54,88,128)(6,16,36,68,109,157,127)(9,19,33,57,91,132,177)(10,20,34,58,92,133,176)(11,21,35,59,93,86,130)(13,30,60,100,147,84,17)(18,38,69,110,158,205,175)(22,42,65,97,137,180,223)(23,43,66,98,138,181,222)(24,44,67,99,139,131,178)(25,45,31,62,52,89,134)(27,55,94,140,50,39,37)(32,61,101,148,195,126,40)(41,71,111,159,206,250,221)(46,75,106,144,185,226,265)(47,76,107,145,186,227,264)(48,77,108,146,187,179,224)(49,78,63,103,90,135,182)(56,95,141,188,83,72,70)(64,102,149,196,241,174,73)(74,113,160,207,251,289,263)(79,117,154,192,231,268,300)(80,118,155,193,232,269,299)(81,119,156,194,233,225,266)(82,120,104,151,136,183,228)(96,142,189,234,125,114,112)(105,150,197,242,282,220,115)(116,162,208,252,290,318,298)(121,166,202,238,273,303,324)(122,167,203,239,274,304,323)(123,168,204,240,275,267,301)(124,169,152,199,184,229,270)(143,190,235,276,173,163,161)(153,198,243,283,314,262,164)(165,210,253,291,319,335,322)(170,214,248,280,308,327,338)(171,215,249,281,309,302,325)(172,216,200,245,230,271,305)(191,236,277,310,219,211,209)(201,244,284,315,333,297,212)(213,255,292,320,336,343,337)(217,258,288,313,330,326,339)(218,259,246,286,272,306,328)(237,278,311,331,261,256,254)(247,285,316,334,342,321,257)(260,295,287,317,307,329,340)(279,312,332,341,296,294,293)(344,345,346,347,348,349,350,351,352,353,354,355,356);(2,6,17,39,45,21,8)(5,13,31,15,16,37,35)(7,18,40,72,78,44,20)(12,27,29,30,59,36,62)(14,32,63,34,38,70,67)(19,41,73,114,120,77,43)(26,52,68,93,60,54,55)(28,56,58,61,99,69,103)(33,64,104,66,71,112,108)(42,74,115,163,169,119,76)(51,86,94,109,88,89,100)(53,90,110,139,101,92,95)(57,96,98,102,146,111,151)(65,105,152,107,113,161,156)(75,116,164,211,216,168,118)(85,128,130,134,140,147,157)(87,131,141,158,133,135,148)(91,136,159,187,149,138,142)(97,143,145,150,194,160,199)(106,153,200,155,162,209,204)(117,165,212,256,259,215,167)(129,176,178,182,188,195,205)(132,179,189,206,181,183,196)(137,184,207,233,197,186,190)(144,191,193,198,240,208,245)(154,201,246,203,210,254,249)(166,213,257,294,295,258,214)(177,222,224,228,234,241,250)(180,225,235,251,227,229,242)(185,230,252,275,243,232,236)(192,237,239,244,281,253,286)(202,247,287,248,255,293,288)(223,264,266,270,276,282,289)(226,267,277,290,269,271,283)(231,272,291,309,284,274,278)(238,279,280,285,313,292,317)(265,299,301,305,310,314,318)(268,302,311,319,304,306,315)(273,307,320,330,316,308,312)(300,323,325,328,331,333,335)(303,326,332,336,327,329,334)(324,338,339,340,341,342,343);(1,3,9,22,46,79,121,4,10,23,47,80,122,170,11,24,48,81,123,171,217,25,49,82,124,172,218,260,50,83,125,173,219,261,296,84,126,174,220,262,297,321,127,175,221,263,298,322,337)(2,7,19,42,75,117,166,8,20,43,76,118,167,214,21,44,77,119,168,215,258,45,78,120,169,216,259,295,39,72,114,163,211,256,294,17,40,73,115,164,212,257,6,18,41,74,116,165,213)(5,14,33,65,106,154,202,15,34,66,107,155,203,248,35,67,108,156,204,249,288,31,63,104,152,200,246,287,37,70,112,161,209,254,293,13,32,64,105,153,201,247,16,38,71,113,162,210,255)(12,28,57,97,144,192,238,29,58,98,145,193,239,280,59,99,146,194,240,281,313,62,103,151,199,245,286,317,27,56,96,143,191,237,279,30,61,102,150,198,244,285,36,69,111,160,208,253,292)(26,53,91,137,185,231,273,54,92,138,186,232,274,308,93,139,187,233,275,309,330,52,90,136,184,230,272,307,55,95,142,190,236,278,312,60,101,149,197,243,284,316,68,110,159,207,252,291,320)(51,87,132,180,226,268,303,88,133,181,227,269,304,327,86,131,179,225,267,302,326,89,135,183,229,271,306,329,94,141,189,235,277,311,332,100,148,196,242,283,315,334,109,158,206,251,290,319,336)(85,129,177,223,265,300,324,128,176,222,264,299,323,338,130,178,224,266,301,325,339,134,182,228,270,305,328,340,140,188,234,276,310,331,341,147,195,241,282,314,333,342,157,205,250,289,318,335,343)5005 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31213, 14);