Finite Group DatabasebetaFinite Group Databasebeta| order | 31200 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| special semilinear | ΣL(2,𝔽25) |
| derived | SL(2,𝔽25) ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, Fitting, socle, hypercenter | C2 | PΣL(2,𝔽25) |
| commutator, cosocle, hypocenter | SL(2,𝔽25) | C2 |
| LCS | G ▷ SL(2,𝔽25) | |
| UCS | C1 ◁ C2 | |
117 conjugacy classes of subgroups; only the 4 normal subgroups are shown.
This group has 117 conjugacy classes of subgroups — too many to draw. Only its 4 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 208 |
| transitive degree | μt(G) = 208 |
A minimal faithful permutation representation — 2 generators on 208 points:
(1,118,17,140,115,14,106,114)(2,119,15,7,175,6,13,98)(3,39,22,107,158,4,44,142)(5,85,86,18,52,91,92,51)(8,129,58,170,124,72,34,101)(9,89,192,88,167,69,189,94)(10,197,23,199,95,21,188,166)(11,196,79,87,82,19,57,93)(12,70,150,26,117,126,168,67)(16,77,157,136,105,116,165,60)(20,173,184,151,182,25,156,201)(24,177,84,64,186,155,122,46)(27,174,183,81,29,37,78,32)(28,42,50,191,41,49,43,31)(30,48,139,169,83,163,62,100)(33,144,198,176,143,112,164,96)(35,153,146,128,56,131,135,71)(36,47,137,104,149,171,59,145)(38,40,127,54,53,147,125,162)(45,138,110,123,74,61,68,90)(55,159,76,73,63,99,121,130)(65,111,207,80,132,195,208,97)(66,206,203,181,205,202,75,161)(102,178,172,133,154,185,179,108)(103,120,193,148,141,134,190,113)(109,200,152,187,194,204,180,160);(1,204,50,208,115,200,43,207)(2,205,15,168,175,66,13,150)(3,22,55,177,158,44,63,155)(4,94,5,71,39,88,52,128)(6,178,25,133,119,185,173,108)(7,59,197,29,98,137,21,27)(8,164,17,47,124,198,106,171)(9,105,135,117,167,16,146,12)(10,153,89,188,95,131,69,23)(11,129,51,65,82,72,18,132)(14,182,169,149,118,20,100,36)(19,157,123,203,196,165,90,75)(24,199,159,201,186,166,99,151)(26,160,61,170,67,187,138,101)(28,183,58,184,41,78,34,156)(30,74,152,85,83,45,180,91)(31,143,48,174,191,33,163,37)(32,141,145,40,81,103,104,147)(35,120,113,192,56,134,148,189)(38,86,127,102,53,92,125,154)(42,109,114,195,49,194,140,111)(46,96,76,190,64,176,121,193)(54,202,112,161,162,206,144,181)(57,179,110,107,79,172,68,142)(60,77,84,73,136,116,122,130)(62,87,80,70,139,93,97,126)28 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((1,118,17,140,115,14,106,114)(2,119,15,7,175,6,13,98)(3,39,22,107,158,4,44,142)(5,85,86,18,52,91,92,51)(8,129,58,170,124,72,34,101)(9,89,192,88,167,69,189,94)(10,197,23,199,95,21,188,166)(11,196,79,87,82,19,57,93)(12,70,150,26,117,126,168,67)(16,77,157,136,105,116,165,60)(20,173,184,151,182,25,156,201)(24,177,84,64,186,155,122,46)(27,174,183,81,29,37,78,32)(28,42,50,191,41,49,43,31)(30,48,139,169,83,163,62,100)(33,144,198,176,143,112,164,96)(35,153,146,128,56,131,135,71)(36,47,137,104,149,171,59,145)(38,40,127,54,53,147,125,162)(45,138,110,123,74,61,68,90)(55,159,76,73,63,99,121,130)(65,111,207,80,132,195,208,97)(66,206,203,181,205,202,75,161)(102,178,172,133,154,185,179,108)(103,120,193,148,141,134,190,113)(109,200,152,187,194,204,180,160), (1,204,50,208,115,200,43,207)(2,205,15,168,175,66,13,150)(3,22,55,177,158,44,63,155)(4,94,5,71,39,88,52,128)(6,178,25,133,119,185,173,108)(7,59,197,29,98,137,21,27)(8,164,17,47,124,198,106,171)(9,105,135,117,167,16,146,12)(10,153,89,188,95,131,69,23)(11,129,51,65,82,72,18,132)(14,182,169,149,118,20,100,36)(19,157,123,203,196,165,90,75)(24,199,159,201,186,166,99,151)(26,160,61,170,67,187,138,101)(28,183,58,184,41,78,34,156)(30,74,152,85,83,45,180,91)(31,143,48,174,191,33,163,37)(32,141,145,40,81,103,104,147)(35,120,113,192,56,134,148,189)(38,86,127,102,53,92,125,154)(42,109,114,195,49,194,140,111)(46,96,76,190,64,176,121,193)(54,202,112,161,162,206,144,181)(57,179,110,107,79,172,68,142)(60,77,84,73,136,116,122,130)(62,87,80,70,139,93,97,126));