Finite Group DatabasebetaFinite Group Databasebeta| order | 14400 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived | Ω+(4,𝔽5) ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, Fitting, socle, hypercenter | C2 | A5×S5 |
| commutator, hypocenter | Ω+(4,𝔽5) | C2 |
| cosocle | SL(2,𝔽5) | C2×A5 |
| LCS | G ▷ Ω+(4,𝔽5) | |
| UCS | C1 ◁ C2 | |
308 conjugacy classes of subgroups; only the 7 normal subgroups are shown.
This group has 308 conjugacy classes of subgroups — too many to draw. Only its 7 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 240 |
| transitive degree | μt(G) = 240 |
A minimal faithful permutation representation — 2 generators on 240 points:
(1,139,32,100,25,119,40,8,198,134,144,126,160,165,29,220,180,73,187,208,2,69,56,159,49,127,16,14,211,177,78,106,101,111,53,205,232,113,195,191)(3,35,21,207,91,105,66,42,99,131,125,130,96,214,94,202,161,34,110,204,9,59,45,190,152,86,129,18,158,176,68,175,156,188,71,224,102,58,79,219)(4,216,23,74,95,22,41,36,163,112,24,84,197,122,44,120,185,173,80,72,10,228,47,140,155,46,17,60,107,166,48,147,217,104,20,97,183,123,133,90)(5,27,212,33,12,37,137,206,154,170,181,193,31,149,196,89,138,30,13,201,11,51,182,57,6,61,64,189,93,117,213,234,55,87,199,151,65,54,7,236)(15,124,239,209,116,178,148,141,215,114,67,76,50,227,43,203,238,172,146,150,39,108,221,192,169,135,85,75,210,167,70,142,26,226,19,237,218,121,83,88)(28,62,92,235,143,225,222,132,164,162,63,184,179,168,145,223,118,229,98,174,52,38,153,200,77,231,240,81,109,103,136,186,233,115,82,194,171,230,157,128);(1,84)(2,147)(3,30)(4,165)(5,126)(6,184)(7,200)(8,96)(9,54)(10,111)(11,106)(12,186)(13,235)(14,156)(15,204)(16,35)(17,69)(18,93)(19,172)(20,34)(21,176)(22,160)(23,166)(24,38)(25,231)(26,230)(27,164)(28,127)(29,113)(31,151)(32,206)(33,218)(36,187)(37,133)(39,219)(40,59)(41,139)(42,154)(43,121)(44,58)(45,131)(46,101)(47,112)(48,62)(49,225)(50,229)(51,109)(52,119)(53,73)(55,89)(56,189)(57,238)(60,195)(61,80)(63,115)(64,88)(65,90)(66,201)(67,223)(68,135)(70,194)(71,132)(72,138)(74,82)(75,211)(76,78)(77,209)(79,149)(81,94)(83,120)(85,228)(86,240)(87,110)(91,175)(92,207)(95,205)(97,146)(98,167)(99,124)(100,171)(102,123)(103,183)(104,182)(105,222)(107,227)(108,158)(114,157)(116,170)(117,169)(118,159)(122,212)(125,178)(128,179)(129,236)(130,152)(134,239)(136,168)(137,150)(140,145)(141,198)(142,144)(143,192)(148,216)(153,190)(155,220)(161,173)(162,185)(163,226)(174,233)(177,221)(180,191)(181,214)(188,213)(193,197)(196,202)(199,224)(203,210)(208,232)(215,237)(217,234)55 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,139,32,100,25,119,40,8,198,134,144,126,160,165,29,220,180,73,187,208,2,69,56,159,49,127,16,14,211,177,78,106,101,111,53,205,232,113,195,191)(3,35,21,207,91,105,66,42,99,131,125,130,96,214,94,202,161,34,110,204,9,59,45,190,152,86,129,18,158,176,68,175,156,188,71,224,102,58,79,219)(4,216,23,74,95,22,41,36,163,112,24,84,197,122,44,120,185,173,80,72,10,228,47,140,155,46,17,60,107,166,48,147,217,104,20,97,183,123,133,90)(5,27,212,33,12,37,137,206,154,170,181,193,31,149,196,89,138,30,13,201,11,51,182,57,6,61,64,189,93,117,213,234,55,87,199,151,65,54,7,236)(15,124,239,209,116,178,148,141,215,114,67,76,50,227,43,203,238,172,146,150,39,108,221,192,169,135,85,75,210,167,70,142,26,226,19,237,218,121,83,88)(28,62,92,235,143,225,222,132,164,162,63,184,179,168,145,223,118,229,98,174,52,38,153,200,77,231,240,81,109,103,136,186,233,115,82,194,171,230,157,128), (1,84)(2,147)(3,30)(4,165)(5,126)(6,184)(7,200)(8,96)(9,54)(10,111)(11,106)(12,186)(13,235)(14,156)(15,204)(16,35)(17,69)(18,93)(19,172)(20,34)(21,176)(22,160)(23,166)(24,38)(25,231)(26,230)(27,164)(28,127)(29,113)(31,151)(32,206)(33,218)(36,187)(37,133)(39,219)(40,59)(41,139)(42,154)(43,121)(44,58)(45,131)(46,101)(47,112)(48,62)(49,225)(50,229)(51,109)(52,119)(53,73)(55,89)(56,189)(57,238)(60,195)(61,80)(63,115)(64,88)(65,90)(66,201)(67,223)(68,135)(70,194)(71,132)(72,138)(74,82)(75,211)(76,78)(77,209)(79,149)(81,94)(83,120)(85,228)(86,240)(87,110)(91,175)(92,207)(95,205)(97,146)(98,167)(99,124)(100,171)(102,123)(103,183)(104,182)(105,222)(107,227)(108,158)(114,157)(116,170)(117,169)(118,159)(122,212)(125,178)(128,179)(129,236)(130,152)(134,239)(136,168)(137,150)(140,145)(141,198)(142,144)(143,192)(148,216)(153,190)(155,220)(161,173)(162,185)(163,226)(174,233)(177,221)(180,191)(181,214)(188,213)(193,197)(196,202)(199,224)(203,210)(208,232)(215,237)(217,234));