Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C53×SD128 ⋊ C2 |
| Sylow | C53 ⋊ #13741 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D32×D53 |
| commutator, cosocle | C32×C53 | C23 |
| Frattini | C32 | C22×D53 |
| Fitting | C53×SD128 | C2 |
| socle | C2×C53 | C2×D32 |
| hypercenter | SD128 | D53 |
| hypocenter | C53 | #13741 |
| LCS | G ▷ C32×C53 ▷ C16×C53 ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ SD128 | |
128 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 128 conjugacy classes of subgroups — too many to draw. Only its 39 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:
(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)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52)(54,57,62,61)(55,64,68,93)(56,70,74,94)(58,75,84,83)(59,76,89,88)(60,77,92,87)(63,95,99,98)(65,100,104,153)(66,101,106,145)(67,90,108,151)(69,109,112,111)(71,113,116,154)(72,114,118,147)(73,91,120,152)(78,121,130,143)(79,122,133,144)(80,123,136,135)(81,124,139,138)(82,125,142,137)(85,126,146,149)(86,127,148,150)(96,155,158,157)(97,156,159,107)(102,160,163,162)(103,128,166,177)(105,161,168,167)(110,169,170,119)(115,131,172,178)(117,171,173,165)(129,140,175,180)(132,141,176,181)(134,174,179,164);(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)(54,56)(55,63)(57,70)(58,79)(59,86)(60,91)(61,94)(62,74)(64,95)(65,102)(66,105)(67,107)(68,99)(69,78)(71,81)(72,80)(73,87)(75,122)(76,127)(77,120)(82,141)(83,144)(84,133)(85,96)(88,150)(89,148)(90,97)(92,152)(93,98)(100,160)(101,161)(103,165)(104,163)(106,168)(108,156)(109,121)(110,140)(111,143)(112,130)(113,124)(114,123)(115,164)(116,139)(117,128)(118,136)(119,129)(125,176)(126,155)(131,134)(132,137)(135,147)(138,154)(142,181)(145,167)(146,158)(149,157)(151,159)(153,162)(166,171)(169,175)(170,180)(172,174)(173,177)(178,179);(54,55)(56,69)(57,64)(58,78)(59,85)(60,90)(61,93)(62,68)(63,71)(65,81)(66,80)(67,87)(70,109)(72,117)(73,119)(74,112)(75,121)(76,126)(77,108)(79,105)(82,140)(83,143)(84,130)(86,102)(88,149)(89,146)(91,110)(92,151)(94,111)(95,113)(96,132)(97,131)(98,154)(99,116)(100,124)(101,123)(103,164)(104,139)(106,136)(107,115)(114,171)(118,173)(120,169)(122,161)(125,175)(127,160)(128,134)(129,137)(133,168)(135,145)(138,153)(141,155)(142,180)(144,167)(147,165)(148,163)(150,162)(152,170)(156,172)(157,181)(158,176)(159,178)(166,174)(177,179)Polycyclic presentation:
2^8,53|c^2=h,d^2=ef,e^2=fg,f^2=gh,g^2=h,b^a=bd,d^a=de,e^a=ef,f^a=fg,g^a=gh,d^b=de,e^b=ef,f^b=fg,g^b=gh,i^b=i^52,i^c=i^52980 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 90621);