Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C79 ⋊ Q256 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #23761612 |
| commutator, cosocle | C64×C79 | C22 |
| Frattini | C64 | C2×D79 |
| Fitting | C79×Q128 | C2 |
| socle, prime radical | C2×C79 | D64 |
| hypercenter | Q128 | D79 |
| hypocenter | C79 | Q256 |
| LCS | G ▷ C64×C79 ▷ C32×C79 ▷ C16×C79 ▷ C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ C64 ◁ Q128 | |
42 conjugacy classes of subgroups.
| minimal degree | μ(G) = 335 |
| transitive degree | μt(G) = 20224 |
A minimal faithful permutation representation — 2 generators on 335 points:
(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)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,82,88,102)(81,89,95,133)(83,104,110,179)(84,112,116,205)(85,118,121,219)(86,123,125,225)(87,127,124,101)(90,135,139,247)(91,141,144,262)(92,146,148,270)(93,150,151,273)(94,152,126,132)(96,153,157,103)(97,134,128,106)(98,140,164,105)(99,145,167,113)(100,149,169,119)(107,188,191,311)(108,193,195,314)(109,197,194,178)(111,129,201,235)(114,209,211,322)(115,213,210,204)(117,130,216,238)(120,221,206,218)(122,131,223,240)(136,249,251,329)(137,253,254,332)(138,255,226,246)(142,264,265,335)(143,266,227,261)(147,271,214,269)(154,278,280,187)(155,281,282,192)(156,174,170,196)(158,248,230,302)(159,252,231,301)(160,186,228,244)(161,275,285,171)(162,263,287,304)(163,183,229,258)(165,276,291,172)(166,207,217,267)(168,277,293,173)(175,242,236,180)(176,241,232,184)(177,272,300,189)(181,203,256,320)(182,202,257,319)(185,224,243,239)(190,313,305,310)(198,260,317,334)(199,259,318,333)(200,274,212,234)(208,233,321,324)(215,268,220,237)(222,245,323,327)(250,331,294,286)(279,308,289,312)(283,330,297,328)(284,295,288,315)(290,296,292,307)(298,303,316,326)(299,306,309,325);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(80,81,88,95)(82,96,102,157)(83,103,110,153)(84,111,116,201)(85,117,121,216)(86,122,125,223)(87,126,124,94)(89,128,133,97)(90,134,139,106)(91,140,144,105)(92,145,148,113)(93,149,151,119)(98,161,164,285)(99,165,167,291)(100,168,169,293)(101,170,127,156)(104,141,179,262)(107,187,191,278)(108,192,195,281)(109,196,194,174)(112,135,205,247)(114,208,211,321)(115,212,210,200)(118,154,219,280)(120,220,206,215)(123,155,225,282)(129,232,235,176)(130,236,238,175)(131,239,240,224)(132,160,152,228)(136,248,251,302)(137,252,254,301)(138,186,226,244)(142,263,265,304)(143,183,227,258)(146,230,270,158)(147,207,214,267)(150,231,273,159)(162,286,287,331)(163,288,229,284)(166,292,217,290)(171,242,275,180)(172,241,276,184)(173,272,277,189)(177,264,300,335)(178,266,197,261)(181,199,256,318)(182,198,257,317)(185,222,243,323)(188,259,311,333)(190,312,305,308)(193,260,314,334)(202,249,319,329)(203,253,320,332)(204,255,213,246)(209,245,322,327)(218,289,221,279)(233,313,324,310)(234,299,274,309)(237,298,268,316)(250,330,294,328)(269,283,271,297)(295,303,315,326)(296,306,307,325)Polycyclic presentation:
2^8,79|a^2=h,b^2=h,c^2=de,d^2=ef,e^2=fg,f^2=gh,g^2=h,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,g^a=gh,c^b=cd,d^b=de,e^b=ef,f^b=fg,g^b=gh,i^b=i^781432 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 57485);