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 | |
305 conjugacy classes of subgroups; only the 7 normal subgroups are shown.
This group has 305 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:
(1,222,7,82,70,135,146,143,196,195,93,182,132,160,213)(2,240,13,145,67,64,83,78,199,187,154,212,65,101,181)(3,52,183,80,239,107,110,96,10,164,48,197,116,68,95)(4,109,24,217,169,137,155,9,28,185,144,218,163,79,156)(5,85,142,49,40,15,98,6,102,234,198,44,56,92,94)(8,108,97,72,89,103,168,105,141,203,170,112,104,200,159)(11,148,77,25,16,39,157,12,161,221,211,20,32,153,71)(14,124,120,90,151,162,115,86,76,238,117,166,122,236,100)(17,55,23,136,180,21,118,63,229,50,19,210,99,29,152)(18,140,231,207,46,150,204,191,123,224,84,125,128,226,139)(22,88,219,208,173,202,147,106,175,233,74,42,75,232,190)(26,43,215,158,53,91,41,31,47,66,225,45,171,134,179)(27,62,167,223,209,57,228,230,214,206,205,30,58,36,186)(33,227,216,188,189,220,54,34,60,184,51,38,114,194,192)(35,113,87,37,69,130,174,119,178,172,177,237,193,165,176)(59,73,149,61,138,81,127,126,133,121,131,201,235,111,129);(1,128,213,62,43,121,48,188,152,84)(2,175,181,38,19,172,24,214,91,147)(3,69,197,124,142,231,154,159,25,237)(4,165,71,117,143,125,5,141,169,130)(6,112,40,178,85,113,92,236,198,139)(7,190,53,216,68,35,23,186,82,75)(8,221,168,110,104,183,72,102,203,240)(9,138,217,108,77,232,93,100,49,201)(10,111,94,170,78,106,11,76,116,81)(12,166,16,133,148,73,153,200,211,74)(13,207,29,230,137,59,47,184,145,140)(14,234,115,79,122,185,90,161,238,222)(15,219,101,220,157,37,80,167,95,22)(17,30,136,233,118,42,50,192,99,193)(18,26,209,158,235,41,54,66,226,171)(20,57,70,208,56,202,109,60,163,174)(21,126,146,89,135,88,31,120,52,129)(27,199,206,180,223,218,58,179,228,187)(28,176,45,119,83,151,64,150,55,97)(32,224,164,36,107,127,44,33,67,191)(34,229,227,195,51,196,189,225,194,239)(39,204,160,205,98,61,144,114,155,46)(63,131,156,105,215,173,65,162,212,149)(86,210,123,132,103,182,87,134,177,96)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((1,222,7,82,70,135,146,143,196,195,93,182,132,160,213)(2,240,13,145,67,64,83,78,199,187,154,212,65,101,181)(3,52,183,80,239,107,110,96,10,164,48,197,116,68,95)(4,109,24,217,169,137,155,9,28,185,144,218,163,79,156)(5,85,142,49,40,15,98,6,102,234,198,44,56,92,94)(8,108,97,72,89,103,168,105,141,203,170,112,104,200,159)(11,148,77,25,16,39,157,12,161,221,211,20,32,153,71)(14,124,120,90,151,162,115,86,76,238,117,166,122,236,100)(17,55,23,136,180,21,118,63,229,50,19,210,99,29,152)(18,140,231,207,46,150,204,191,123,224,84,125,128,226,139)(22,88,219,208,173,202,147,106,175,233,74,42,75,232,190)(26,43,215,158,53,91,41,31,47,66,225,45,171,134,179)(27,62,167,223,209,57,228,230,214,206,205,30,58,36,186)(33,227,216,188,189,220,54,34,60,184,51,38,114,194,192)(35,113,87,37,69,130,174,119,178,172,177,237,193,165,176)(59,73,149,61,138,81,127,126,133,121,131,201,235,111,129), (1,128,213,62,43,121,48,188,152,84)(2,175,181,38,19,172,24,214,91,147)(3,69,197,124,142,231,154,159,25,237)(4,165,71,117,143,125,5,141,169,130)(6,112,40,178,85,113,92,236,198,139)(7,190,53,216,68,35,23,186,82,75)(8,221,168,110,104,183,72,102,203,240)(9,138,217,108,77,232,93,100,49,201)(10,111,94,170,78,106,11,76,116,81)(12,166,16,133,148,73,153,200,211,74)(13,207,29,230,137,59,47,184,145,140)(14,234,115,79,122,185,90,161,238,222)(15,219,101,220,157,37,80,167,95,22)(17,30,136,233,118,42,50,192,99,193)(18,26,209,158,235,41,54,66,226,171)(20,57,70,208,56,202,109,60,163,174)(21,126,146,89,135,88,31,120,52,129)(27,199,206,180,223,218,58,179,228,187)(28,176,45,119,83,151,64,150,55,97)(32,224,164,36,107,127,44,33,67,191)(34,229,227,195,51,196,189,225,194,239)(39,204,160,205,98,61,144,114,155,46)(63,131,156,105,215,173,65,162,212,149)(86,210,123,132,103,182,87,134,177,96));