Finite Group DatabasebetaFinite Group Databasebeta| order | 18176 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C71 · Schur multiplier (H2) = C4 |
| direct product | C71×#13106 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C71 | C2×D4 |
| commutator | C2×C4 | C22×C8×C71 |
| Frattini, cosocle | C22×C8 | C23×C71 |
| socle | C22×C71 | C8×D4 |
| prime radical | C23×C71 | #131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C71 ◁ C22×C16×C71 ◁ G | |
272 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 4544 |
A minimal faithful permutation representation — 3 generators on 111 points:
(1,2)(3,6)(4,5)(7,8)(9,10,15,20,14,21,31,38,13,18,30,22,29,19,36,37)(11,12,25,27,24,28,35,40,23,16,32,33,39,26,17,34)(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);(2,5)(6,8)(9,11,15,25,14,24,31,35,13,23,30,32,29,39,36,17)(10,16,20,33,21,26,38,34,18,12,22,27,19,28,37,40);(1,3,4,7)(2,6,5,8)(9,12,13,16)(10,17,18,35)(11,22,23,20)(14,28,29,26)(15,27,30,33)(19,32,21,25)(24,37,39,38)(31,40,36,34)5396 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18176, 6094);