Finite Group DatabasebetaFinite Group Databasebeta| order | 30848 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
| dicyclic | Dic7712 |
| Sylow | C241 ⋊ Q128 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D7712 |
| commutator, cosocle | C32×C241 | C22 |
| Frattini | C32 | C2×D241 |
| Fitting | C64×C241 | C2 |
| socle, prime radical | C2×C241 | D32 |
| hypercenter | C64 | D241 |
| hypocenter | C241 | Q128 |
| LCS | G ▷ C32×C241 ▷ C16×C241 ▷ C8×C241 ▷ C4×C241 ▷ C2×C241 ▷ C241 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ C64 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 369 |
| transitive degree | μt(G) = 30848 |
A minimal faithful permutation representation — 2 generators on 369 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,128,127,126,125,124,123,122,121,120,119,118,117,116,115,114,113,112,111,110,109,108,107,106,105,104,103,102,101,100,99,98,97,96,95,94,93,92,91,90,89,88,87,86,85,84,83,82,81,80,79,78,77,76,75,74,73,72,71,70,69,68,67,66)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369);(1,65,33,97)(2,66,34,98)(3,67,35,99)(4,68,36,100)(5,69,37,101)(6,70,38,102)(7,71,39,103)(8,72,40,104)(9,73,41,105)(10,74,42,106)(11,75,43,107)(12,76,44,108)(13,77,45,109)(14,78,46,110)(15,79,47,111)(16,80,48,112)(17,81,49,113)(18,82,50,114)(19,83,51,115)(20,84,52,116)(21,85,53,117)(22,86,54,118)(23,87,55,119)(24,88,56,120)(25,89,57,121)(26,90,58,122)(27,91,59,123)(28,92,60,124)(29,93,61,125)(30,94,62,126)(31,95,63,127)(32,96,64,128)(130,369)(131,368)(132,367)(133,366)(134,365)(135,364)(136,363)(137,362)(138,361)(139,360)(140,359)(141,358)(142,357)(143,356)(144,355)(145,354)(146,353)(147,352)(148,351)(149,350)(150,349)(151,348)(152,347)(153,346)(154,345)(155,344)(156,343)(157,342)(158,341)(159,340)(160,339)(161,338)(162,337)(163,336)(164,335)(165,334)(166,333)(167,332)(168,331)(169,330)(170,329)(171,328)(172,327)(173,326)(174,325)(175,324)(176,323)(177,322)(178,321)(179,320)(180,319)(181,318)(182,317)(183,316)(184,315)(185,314)(186,313)(187,312)(188,311)(189,310)(190,309)(191,308)(192,307)(193,306)(194,305)(195,304)(196,303)(197,302)(198,301)(199,300)(200,299)(201,298)(202,297)(203,296)(204,295)(205,294)(206,293)(207,292)(208,291)(209,290)(210,289)(211,288)(212,287)(213,286)(214,285)(215,284)(216,283)(217,282)(218,281)(219,280)(220,279)(221,278)(222,277)(223,276)(224,275)(225,274)(226,273)(227,272)(228,271)(229,270)(230,269)(231,268)(232,267)(233,266)(234,265)(235,264)(236,263)(237,262)(238,261)(239,260)(240,259)(241,258)(242,257)(243,256)(244,255)(245,254)(246,253)(247,252)(248,251)(249,250)Polycyclic presentation:
2^7,241|a^2=g,b^2=g,c^2=de,d^2=ef,e^2=fg,f^2=g,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,h^a=h^240,c^b=cd,d^b=de,e^b=ef,f^b=fg,h^b=h^2407715 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30848, 2728);