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, Fitting, hypercenter | C4 | A52 |
| commutator, hypocenter | Ω+(4,𝔽5) | C2 |
| Frattini, socle, cosocle | C2 | C2×A52 |
| LCS | G ▷ Ω+(4,𝔽5) | |
| UCS | C1 ◁ C4 | |
445 conjugacy classes of subgroups; only the 9 normal subgroups are shown.
This group has 445 conjugacy classes of subgroups — too many to draw. Only its 9 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 144 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 144 points:
(1,25,106,7,18,52,2,42,67,12,35,82)(3,115,103,16,128,76,8,85,64,33,129,110)(4,70,97,20,26,134,9,80,56,37,43,122)(5,74,83,24,87,143,10,109,114,41,117,138)(6,141,72,29,66,27,11,124,84,46,105,44)(13,21,95,49,71,123,30,38,54,92,108,126)(14,96,32,53,77,63,31,55,15,94,111,57)(17,133,39,23,119,104,34,135,22,40,130,65)(19,48,62,58,86,99,36,91,102,98,116,59)(28,136,125,89,107,93,45,144,142,118,69,51)(47,79,131,120,61,112,90,75,137,127,101,78)(50,100,140,81,68,132,88,60,139,113,73,121);(1,52,9,70,119,2,82,4,80,130)(3,44,71,32,59,8,27,108,15,99)(5,123,23,11,50,10,126,40,6,88)(7,67,37,26,135,12,106,20,43,133)(13,64,58,77,141,30,103,98,111,124)(14,101,138,60,25,31,61,143,100,42)(16,72,38,63,62,33,84,21,57,102)(17,29,113,24,54,34,46,81,41,95)(18,94,75,114,68,35,53,79,83,73)(19,96,105,49,110,36,55,66,92,76)(22,107,93,90,112,39,69,51,47,78)(28,115,122,117,140,45,85,134,87,139)(48,91)(56,109,121,89,128,97,74,132,118,129)(65,144,142,127,137,104,136,125,120,131)(86,116)82 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((1,25,106,7,18,52,2,42,67,12,35,82)(3,115,103,16,128,76,8,85,64,33,129,110)(4,70,97,20,26,134,9,80,56,37,43,122)(5,74,83,24,87,143,10,109,114,41,117,138)(6,141,72,29,66,27,11,124,84,46,105,44)(13,21,95,49,71,123,30,38,54,92,108,126)(14,96,32,53,77,63,31,55,15,94,111,57)(17,133,39,23,119,104,34,135,22,40,130,65)(19,48,62,58,86,99,36,91,102,98,116,59)(28,136,125,89,107,93,45,144,142,118,69,51)(47,79,131,120,61,112,90,75,137,127,101,78)(50,100,140,81,68,132,88,60,139,113,73,121), (1,52,9,70,119,2,82,4,80,130)(3,44,71,32,59,8,27,108,15,99)(5,123,23,11,50,10,126,40,6,88)(7,67,37,26,135,12,106,20,43,133)(13,64,58,77,141,30,103,98,111,124)(14,101,138,60,25,31,61,143,100,42)(16,72,38,63,62,33,84,21,57,102)(17,29,113,24,54,34,46,81,41,95)(18,94,75,114,68,35,53,79,83,73)(19,96,105,49,110,36,55,66,92,76)(22,107,93,90,112,39,69,51,47,78)(28,115,122,117,140,45,85,134,87,139)(48,91)(56,109,121,89,128,97,74,132,118,129)(65,144,142,127,137,104,136,125,120,131)(86,116));