Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C256 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | C53 ⋊ C256 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C128 | D53 |
| commutator, hypocenter | C53 | C256 |
| Fitting, cosocle | C53×C128 | C2 |
| socle, prime radical | C2×C53 | C128 |
| LCS | G ▷ C53 | |
| UCS | C1 ◁ C128 | |
18 conjugacy classes of subgroups.
| minimal degree | μ(G) = 309 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 2 generators on 309 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)(54,55,56,63,57,64,70,91,58,65,71,92,76,97,112,147,59,66,72,93,77,98,113,148,81,102,117,152,127,162,182,217,60,67,73,94,78,99,114,149,82,103,118,153,128,163,183,218,85,106,121,156,131,166,186,221,137,172,192,227,202,237,252,273,61,68,74,95,79,100,115,150,83,104,119,154,129,164,184,219,86,107,122,157,132,167,187,222,138,173,193,228,203,238,253,274,88,109,124,159,134,169,189,224,140,175,195,230,205,240,255,276,143,178,198,233,208,243,258,279,212,247,262,283,267,288,294,301,62,69,75,96,80,101,116,151,84,105,120,155,130,165,185,220,87,108,123,158,133,168,188,223,139,174,194,229,204,239,254,275,89,110,125,160,135,170,190,225,141,176,196,231,206,241,256,277,144,179,199,234,209,244,259,280,213,248,263,284,268,289,295,302,90,111,126,161,136,171,191,226,142,177,197,232,207,242,257,278,145,180,200,235,210,245,260,281,214,249,264,285,269,290,296,303,146,181,201,236,211,246,261,282,215,250,265,286,270,291,297,304,216,251,266,287,271,292,298,305,272,293,299,306,300,307,308,309);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)Polycyclic presentation:
2^8,53|a^2=b,b^2=c,c^2=d,d^2=e,e^2=f,f^2=g,g^2=h,i^a=i^523584 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 56093);