Finite Group DatabasebetaFinite Group Databasebeta| order | 20160 |
| isoclinism | isoclinic class of #72209753 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, Fitting, socle, cosocle, hypercenter | C2 | A5×GL(3,𝔽2) |
| commutator, hypocenter | G | C1 |
| LCS | G | |
| UCS | C1 ◁ C2 | |
221 conjugacy classes of subgroups; only the 5 normal subgroups are shown.
This group has 221 conjugacy classes of subgroups — too many to draw. Only its 5 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Permutation generators:
(1,2,8,3)(4,14,12,15)(5,9,6,10)(7,11,19,13)(16,27,17,28)(18,29,24,32)(20,22,21,23)(25,43,30,44)(26,48,31,49)(33,53,34,54)(35,39,37,41)(36,40,38,42)(45,70,46,71)(47,72,55,74)(50,77,51,78)(52,79,56,80)(57,83,59,85)(58,84,60,86)(61,65,62,66)(63,67,64,68)(69,91,73,92)(75,96,76,97)(81,103,82,104)(87,109,88,110)(89,111,90,112)(93,115,94,116)(95,117,98,120)(99,123,101,125)(100,124,102,126)(105,131,107,133)(106,132,108,134)(113,119,118,114)(121,145,122,146)(127,153,128,154)(129,155,130,156)(135,157,136,158)(137,159,138,160)(139,143,140,142)(141,148,147,144)(149,165,151,167)(150,166,152,168)(161,164,162,163)(169,181,170,182)(171,183,172,184)(173,179,175,177)(174,180,176,178)(185,190,186,189)(187,192,188,191);(1,4,2)(3,8,12)(5,16,9)(6,17,10)(7,18,11)(13,19,24)(14,25,26)(15,30,31)(20,33,22)(21,34,23)(27,45,50)(28,46,51)(29,47,52)(32,55,56)(35,57,39)(36,58,40)(37,59,41)(38,60,42)(43,49,69)(44,48,73)(53,75,81)(54,76,82)(61,87,65)(62,88,66)(63,89,67)(64,90,68)(70,78,93)(71,77,94)(72,80,95)(74,79,98)(83,99,105)(84,100,106)(85,101,107)(86,102,108)(91,113,114)(92,118,119)(96,104,121)(97,103,122)(109,127,135)(110,128,136)(111,129,137)(112,130,138)(115,139,142)(116,140,143)(117,141,144)(120,147,148)(123,133,149)(124,134,150)(125,131,151)(126,132,152)(145,161,163)(146,162,164)(153,158,169)(154,157,170)(155,160,171)(156,159,172)(165,173,177)(166,174,178)(167,175,179)(168,176,180)(181,185,189)(182,186,190)(183,187,191)(184,188,192);(1,5,8,6)(2,9,3,10)(4,16,12,17)(7,20,19,21)(11,22,13,23)(14,27,15,28)(18,33,24,34)(25,45,30,46)(26,50,31,51)(29,53,32,54)(35,61,37,62)(36,63,38,64)(39,65,41,66)(40,67,42,68)(43,70,44,71)(47,75,55,76)(48,77,49,78)(52,81,56,82)(57,87,59,88)(58,89,60,90)(69,93,73,94)(72,96,74,97)(79,103,80,104)(83,109,85,110)(84,111,86,112)(91,115,92,116)(95,121,98,122)(99,127,101,128)(100,129,102,130)(105,135,107,136)(106,137,108,138)(113,139,118,140)(114,142,119,143)(117,145,120,146)(123,153,125,154)(124,155,126,156)(131,157,133,158)(132,159,134,160)(141,161,147,162)(144,163,148,164)(149,169,151,170)(150,171,152,172)(165,181,167,182)(166,183,168,184)(173,185,175,186)(174,187,176,188)(177,189,179,190)(178,191,180,192);(1,7,5)(2,11,9)(3,13,10)(4,18,16)(6,8,19)(12,24,17)(14,29,27)(15,32,28)(20,35,36)(21,37,38)(22,39,40)(23,41,42)(25,47,45)(26,52,50)(30,55,46)(31,56,51)(33,57,58)(34,59,60)(43,72,70)(44,74,71)(48,79,77)(49,80,78)(53,83,84)(54,85,86)(69,95,93)(73,98,94)(75,99,100)(76,101,102)(81,105,106)(82,107,108)(91,117,115)(92,120,116)(96,123,124)(97,125,126)(103,131,132)(104,133,134)(113,141,139)(114,144,142)(118,147,140)(119,148,143)(121,149,150)(122,151,152)(145,165,166)(146,167,168)(161,173,174)(162,175,176)(163,177,178)(164,179,180)50 conjugacy classes — view the full character table.
| maximal quotient of | SL(2,𝔽5)×SL(2,𝔽7) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:
gap> G := Group((1,2,8,3)(4,14,12,15)(5,9,6,10)(7,11,19,13)(16,27,17,28)(18,29,24,32)(20,22,21,23)(25,43,30,44)(26,48,31,49)(33,53,34,54)(35,39,37,41)(36,40,38,42)(45,70,46,71)(47,72,55,74)(50,77,51,78)(52,79,56,80)(57,83,59,85)(58,84,60,86)(61,65,62,66)(63,67,64,68)(69,91,73,92)(75,96,76,97)(81,103,82,104)(87,109,88,110)(89,111,90,112)(93,115,94,116)(95,117,98,120)(99,123,101,125)(100,124,102,126)(105,131,107,133)(106,132,108,134)(113,119,118,114)(121,145,122,146)(127,153,128,154)(129,155,130,156)(135,157,136,158)(137,159,138,160)(139,143,140,142)(141,148,147,144)(149,165,151,167)(150,166,152,168)(161,164,162,163)(169,181,170,182)(171,183,172,184)(173,179,175,177)(174,180,176,178)(185,190,186,189)(187,192,188,191), (1,4,2)(3,8,12)(5,16,9)(6,17,10)(7,18,11)(13,19,24)(14,25,26)(15,30,31)(20,33,22)(21,34,23)(27,45,50)(28,46,51)(29,47,52)(32,55,56)(35,57,39)(36,58,40)(37,59,41)(38,60,42)(43,49,69)(44,48,73)(53,75,81)(54,76,82)(61,87,65)(62,88,66)(63,89,67)(64,90,68)(70,78,93)(71,77,94)(72,80,95)(74,79,98)(83,99,105)(84,100,106)(85,101,107)(86,102,108)(91,113,114)(92,118,119)(96,104,121)(97,103,122)(109,127,135)(110,128,136)(111,129,137)(112,130,138)(115,139,142)(116,140,143)(117,141,144)(120,147,148)(123,133,149)(124,134,150)(125,131,151)(126,132,152)(145,161,163)(146,162,164)(153,158,169)(154,157,170)(155,160,171)(156,159,172)(165,173,177)(166,174,178)(167,175,179)(168,176,180)(181,185,189)(182,186,190)(183,187,191)(184,188,192), (1,5,8,6)(2,9,3,10)(4,16,12,17)(7,20,19,21)(11,22,13,23)(14,27,15,28)(18,33,24,34)(25,45,30,46)(26,50,31,51)(29,53,32,54)(35,61,37,62)(36,63,38,64)(39,65,41,66)(40,67,42,68)(43,70,44,71)(47,75,55,76)(48,77,49,78)(52,81,56,82)(57,87,59,88)(58,89,60,90)(69,93,73,94)(72,96,74,97)(79,103,80,104)(83,109,85,110)(84,111,86,112)(91,115,92,116)(95,121,98,122)(99,127,101,128)(100,129,102,130)(105,135,107,136)(106,137,108,138)(113,139,118,140)(114,142,119,143)(117,145,120,146)(123,153,125,154)(124,155,126,156)(131,157,133,158)(132,159,134,160)(141,161,147,162)(144,163,148,164)(149,169,151,170)(150,171,152,172)(165,181,167,182)(166,183,168,184)(173,185,175,186)(174,187,176,188)(177,189,179,190)(178,191,180,192), (1,7,5)(2,11,9)(3,13,10)(4,18,16)(6,8,19)(12,24,17)(14,29,27)(15,32,28)(20,35,36)(21,37,38)(22,39,40)(23,41,42)(25,47,45)(26,52,50)(30,55,46)(31,56,51)(33,57,58)(34,59,60)(43,72,70)(44,74,71)(48,79,77)(49,80,78)(53,83,84)(54,85,86)(69,95,93)(73,98,94)(75,99,100)(76,101,102)(81,105,106)(82,107,108)(91,117,115)(92,120,116)(96,123,124)(97,125,126)(103,131,132)(104,133,134)(113,141,139)(114,144,142)(118,147,140)(119,148,143)(121,149,150)(122,151,152)(145,165,166)(146,167,168)(161,173,174)(162,175,176)(163,177,178)(164,179,180));