Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C22×C3 |
| direct product | C2×(#131⋊C3)⋊A4 |
| Fitting, Sylow | C2×#3573 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C8 | A42 |
| commutator, hypocenter | H(5,𝔽2) | C2×C32×C4 |
| Frattini | C4 | C22×A42 |
| Fitting | C2×#3573 | C32 |
| socle | C22 | C4×A42 |
| prime radical | C2×(#41⋊C3)⋊A4 | C2 |
| cosocle | #585 | C22×C32 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C2×C8 | |
1086 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 1086 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 98 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 2 generators on 98 points:
(3,43,62,39,28,44,24,77,29,14,59,38,9,41,23,11,26,13,6,79,8,4,61,80)(5,73,52,72,55,34,54,76,19,36,58,69,21,40,51,31,25,33,16,94,18,10,86,70)(7,42,22,78,27,12,60,37)(15,75,97,92,57,93,84,91,85,67,83,90,49,66,82,63,48,65,45,96,47,30,88,98)(17,35,50,71,20,32,53,68)(46,74,81,95,56,64,87,89);(1,2)(3,20,49,8,5,57,9,17,15,29,21,48)(4,35,67,13,10,75,14,32,30,44,36,66)(6,53,84,23,16,88,24,50,45,62,54,83)(7,18,85,28,25,46,27,19,47,26,55,56)(11,71,92,38,31,96,39,68,63,80,72,91)(12,33,93,43,40,64,42,34,65,41,73,74)(22,51,97,61,58,81,60,52,82,59,86,87)(37,69,98,79,76,89,78,70,90,77,94,95)200 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72228092, #72253041, #72253043, … |
| maximal normal subgroup of | C2×#72228092, #72253041, #72253043, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,43,62,39,28,44,24,77,29,14,59,38,9,41,23,11,26,13,6,79,8,4,61,80)(5,73,52,72,55,34,54,76,19,36,58,69,21,40,51,31,25,33,16,94,18,10,86,70)(7,42,22,78,27,12,60,37)(15,75,97,92,57,93,84,91,85,67,83,90,49,66,82,63,48,65,45,96,47,30,88,98)(17,35,50,71,20,32,53,68)(46,74,81,95,56,64,87,89), (1,2)(3,20,49,8,5,57,9,17,15,29,21,48)(4,35,67,13,10,75,14,32,30,44,36,66)(6,53,84,23,16,88,24,50,45,62,54,83)(7,18,85,28,25,46,27,19,47,26,55,56)(11,71,92,38,31,96,39,68,63,80,72,91)(12,33,93,43,40,64,42,34,65,41,73,74)(22,51,97,61,58,81,60,52,82,59,86,87)(37,69,98,79,76,89,78,70,90,77,94,95));