Finite Group DatabasebetaFinite Group Databasebeta| order | 23872 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
| dicyclic | Dic5968 |
| Sylow | C373 ⋊ Q64 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D5968 |
| commutator, cosocle | C16×C373 | C22 |
| Frattini | C16 | C2×D373 |
| Fitting | C32×C373 | C2 |
| socle, prime radical | C2×C373 | D16 |
| hypercenter | C32 | D373 |
| hypocenter | C373 | Q64 |
| LCS | G ▷ C16×C373 ▷ C8×C373 ▷ C4×C373 ▷ C2×C373 ▷ C373 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 437 |
| transitive degree | μt(G) = 23872 |
A minimal faithful permutation representation — 2 generators on 437 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)(33,64,63,62,61,60,59,58,57,56,55,54,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437);(1,33,17,49)(2,34,18,50)(3,35,19,51)(4,36,20,52)(5,37,21,53)(6,38,22,54)(7,39,23,55)(8,40,24,56)(9,41,25,57)(10,42,26,58)(11,43,27,59)(12,44,28,60)(13,45,29,61)(14,46,30,62)(15,47,31,63)(16,48,32,64)(66,437)(67,436)(68,435)(69,434)(70,433)(71,432)(72,431)(73,430)(74,429)(75,428)(76,427)(77,426)(78,425)(79,424)(80,423)(81,422)(82,421)(83,420)(84,419)(85,418)(86,417)(87,416)(88,415)(89,414)(90,413)(91,412)(92,411)(93,410)(94,409)(95,408)(96,407)(97,406)(98,405)(99,404)(100,403)(101,402)(102,401)(103,400)(104,399)(105,398)(106,397)(107,396)(108,395)(109,394)(110,393)(111,392)(112,391)(113,390)(114,389)(115,388)(116,387)(117,386)(118,385)(119,384)(120,383)(121,382)(122,381)(123,380)(124,379)(125,378)(126,377)(127,376)(128,375)(129,374)(130,373)(131,372)(132,371)(133,370)(134,369)(135,368)(136,367)(137,366)(138,365)(139,364)(140,363)(141,362)(142,361)(143,360)(144,359)(145,358)(146,357)(147,356)(148,355)(149,354)(150,353)(151,352)(152,351)(153,350)(154,349)(155,348)(156,347)(157,346)(158,345)(159,344)(160,343)(161,342)(162,341)(163,340)(164,339)(165,338)(166,337)(167,336)(168,335)(169,334)(170,333)(171,332)(172,331)(173,330)(174,329)(175,328)(176,327)(177,326)(178,325)(179,324)(180,323)(181,322)(182,321)(183,320)(184,319)(185,318)(186,317)(187,316)(188,315)(189,314)(190,313)(191,312)(192,311)(193,310)(194,309)(195,308)(196,307)(197,306)(198,305)(199,304)(200,303)(201,302)(202,301)(203,300)(204,299)(205,298)(206,297)(207,296)(208,295)(209,294)(210,293)(211,292)(212,291)(213,290)(214,289)(215,288)(216,287)(217,286)(218,285)(219,284)(220,283)(221,282)(222,281)(223,280)(224,279)(225,278)(226,277)(227,276)(228,275)(229,274)(230,273)(231,272)(232,271)(233,270)(234,269)(235,268)(236,267)(237,266)(238,265)(239,264)(240,263)(241,262)(242,261)(243,260)(244,259)(245,258)(246,257)(247,256)(248,255)(249,254)(250,253)(251,252)Polycyclic presentation:
2^6,373|a^2=f,b^2=f,c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^372,c^b=cd,d^b=de,e^b=ef,g^b=g^3725971 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(23872, 390);