Finite Group DatabasebetaFinite Group Databasebeta| order | 17536 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C137 · Schur multiplier (H2) = C23 |
| direct product | #2138×C137 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C137 | D8 |
| commutator | C8 | C22×C4×C137 |
| Frattini, cosocle | C2×C8 | C23×C137 |
| socle | C22×C137 | C2×D8 |
| prime radical | C2×#135×C137 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C137 ◁ C22×C4×C137 ◁ C22×C8×C137 ◁ G | |
228 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 173 |
| transitive degree | μt(G) = 8768 |
A minimal faithful permutation representation — 3 generators on 173 points:
(1,2,3,4)(5,6)(7,16)(8,13)(9,21)(10,25)(11,15)(12,18)(14,23)(17,32)(19,33)(20,31)(22,27)(24,35)(26,30)(28,36)(29,34)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173);(5,7)(6,12)(8,17)(9,22)(10,26)(11,19)(13,27)(14,30)(15,29)(16,21)(18,23)(20,34)(24,36)(25,28)(31,32)(33,35);(5,8,11,10)(6,13,15,25)(7,17,19,26)(9,20,24,23)(12,27,29,28)(14,21,31,35)(16,32,33,30)(18,22,34,36)6028 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C137×#12641, C137×#12642, C137×#12745, … |
| maximal normal subgroup of | C137×#12641, C137×#12642, C137×#12745, … |
| maximal quotient of | C137×#11591, C137×#11647, C137×#11659, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17536, 870);