Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C53×#2264 ⋊ C2 |
| Sylow | C53 ⋊ #13672 |
274 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 55 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 — 3 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)(54,57,61,80)(55,63,67,101)(56,69,74,117)(58,76,86,133)(59,79,88,77)(60,78,91,136)(62,93,97,152)(64,98,105,159)(65,100,87,75)(66,99,108,161)(68,109,112,166)(70,114,92,149)(71,120,73,115)(72,116,123,172)(81,137,140,174)(82,128,143,177)(83,129,106,134)(84,132,146,130)(85,131,102,162)(89,148,147,126)(90,135,121,173)(94,153,96,124)(95,151,155,168)(103,157,165,171)(104,158,139,125)(107,160,154,180)(110,167,111,113)(118,150,119,170)(122,169,164,181)(127,138,175,145)(141,179,142,176)(144,156,163,178);(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)(54,56)(55,62)(57,69)(58,82)(59,71)(60,90)(61,74)(63,93)(64,103)(65,94)(66,107)(67,97)(68,81)(70,85)(72,83)(73,88)(75,124)(76,128)(77,115)(78,135)(79,120)(80,117)(84,141)(86,143)(87,96)(89,95)(91,121)(92,102)(98,157)(99,160)(100,153)(101,152)(104,164)(105,165)(106,123)(108,154)(109,137)(110,138)(111,145)(112,140)(113,127)(114,131)(116,129)(118,144)(119,163)(122,139)(125,169)(126,168)(130,176)(132,179)(133,177)(134,172)(136,173)(142,146)(147,155)(148,151)(149,162)(150,156)(158,181)(159,171)(161,180)(166,174)(167,175)(170,178);(54,55,61,67)(56,68,74,112)(57,75,80,100)(58,81,86,140)(59,87,88,65)(60,89,91,147)(62,92,97,70)(63,79,101,77)(64,102,105,85)(66,106,108,83)(69,113,117,167)(71,111,73,110)(72,122,123,164)(76,127,133,175)(78,99,136,161)(82,107,143,154)(84,145,146,138)(90,103,121,165)(93,150,152,170)(94,118,96,119)(95,141,155,142)(98,156,159,178)(104,144,139,163)(109,120,166,115)(114,124,149,153)(116,171,172,157)(125,162,158,131)(126,134,148,129)(128,168,177,151)(130,137,132,174)(135,169,173,181)(160,179,180,176)Polycyclic presentation:
2^8,53|a^2=h,c^2=h,d^2=fg,e^2=h,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fg,g^a=gh,d^b=df,f^b=fg,g^b=gh,e^c=eh,i^c=i^52713 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 90270);