Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×C53×Q64 ⋊ C2 |
| Sylow | C53 ⋊ #33984 |
452 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 452 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 181 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 181 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)(57,74)(63,95)(69,111)(75,119)(76,120)(77,121)(91,143)(96,147)(97,148)(98,129)(108,160)(112,142)(113,162)(114,130)(122,159)(123,166)(124,152)(125,149)(126,167)(127,168)(128,169)(144,177)(145,154)(150,178)(151,170)(153,179)(161,165)(163,172)(164,181)(171,175)(173,176)(174,180);(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)(54,57)(55,63)(56,69)(58,75)(59,76)(60,77)(61,74)(62,91)(64,96)(65,97)(66,98)(67,95)(68,108)(70,112)(71,113)(72,114)(73,111)(78,122)(79,123)(80,124)(81,125)(82,126)(83,119)(84,127)(85,128)(86,121)(87,120)(88,129)(89,130)(90,142)(92,144)(93,145)(94,143)(99,149)(100,150)(101,151)(102,147)(103,152)(104,153)(105,148)(106,154)(107,159)(109,161)(110,160)(115,163)(116,164)(117,162)(118,165)(131,170)(132,171)(133,172)(134,173)(135,166)(136,174)(137,167)(138,175)(139,176)(140,168)(141,169)(146,177)(155,180)(156,178)(157,181)(158,179);(54,56,61,73)(55,62,67,94)(57,69,74,111)(58,79,83,135)(59,85,87,141)(60,89,86,72)(63,91,95,143)(64,100,102,156)(65,104,105,158)(66,106,88,93)(68,107,110,78)(70,99,90,81)(71,103,117,80)(75,123,119,166)(76,128,120,169)(77,130,121,114)(82,139,137,134)(84,92,140,146)(96,150,147,178)(97,153,148,179)(98,154,129,145)(101,157,131,116)(108,159,160,122)(109,132,118,138)(112,149,142,125)(113,152,162,124)(115,136,133,155)(126,176,167,173)(127,144,168,177)(151,181,170,164)(161,171,165,175)(163,174,172,180);(54,55,61,67)(56,68,73,110)(57,63,74,95)(58,78,83,107)(59,84,87,140)(60,88,86,66)(62,90,94,70)(64,99,102,81)(65,103,105,80)(69,108,111,160)(71,116,117,157)(72,118,89,109)(75,122,119,159)(76,127,120,168)(77,129,121,98)(79,104,135,158)(82,138,137,132)(85,100,141,156)(91,142,143,112)(92,139,146,134)(93,115,106,133)(96,149,147,125)(97,152,148,124)(101,136,131,155)(113,164,162,181)(114,165,130,161)(123,153,166,179)(126,175,167,171)(128,150,169,178)(144,176,177,173)(145,163,154,172)(151,174,170,180)Polycyclic presentation:
2^8,53|a^2=h,b^2=h,e^2=fg,f^2=gh,g^2=h,b^a=be,e^a=ef,f^a=fg,g^a=gh,e^b=ef,f^b=fg,g^b=gh,d^c=dh,i^d=i^521040 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 327727);