Finite Group DatabasebetaFinite Group Databasebeta| order | 17792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C139 · Schur multiplier (H2) = C24 |
| direct product | C2×#327×C139 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C139 | C22⋊C4 |
| commutator | C2×C4 | C22×C4×C139 |
| Frattini, cosocle | C22×C4 | C23×C139 |
| prime radical | C2×#121×C139 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C139 ◁ C23×C4×C139 ◁ G | |
388 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 388 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 157 |
| transitive degree | μt(G) = 4448 |
A minimal faithful permutation representation — 3 generators on 157 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157);(3,4)(5,9,10,7)(6,8)(11,12)(13,15)(14,17)(16,18);(3,5)(4,7)(6,10)(8,9)(12,15)(13,18)(16,17)5282 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C4×#327×C139, C139×#8216, C139×#8218, … |
| maximal normal subgroup of | C4×#327×C139, C139×#8216, C139×#8218, … |
| maximal quotient of | C2×#1270×C139, C2×#1271×C139, C139×#7565, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17792, 230);