Finite Group DatabasebetaFinite Group Databasebeta| order | 17536 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C137 · Schur multiplier (H2) = C4 |
| direct product | #2164×C137 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C137 | C23 |
| commutator | C22 | C22×C8×C137 |
| Frattini, cosocle | C2×C8 | C23×C137 |
| socle | C22×C137 | C22×C8 |
| prime radical | C23×C137 | C2×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C137 ◁ G | |
138 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 138 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 177 |
| transitive degree | μt(G) = 8768 |
A minimal faithful permutation representation — 3 generators on 177 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,16,13,17,22,28,14,18,23,29,24,30,34,38)(11,15,20,26,21,27,33,37,19,25,31,35,32,36,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177);(2,5)(6,8)(9,11)(10,15)(12,20)(13,21)(14,19)(16,26)(17,27)(18,25)(22,33)(23,31)(24,32)(28,37)(29,35)(30,36)(34,39)(38,40);(1,3,4,7)(2,6,5,8)(11,19)(15,25)(20,31)(21,32)(26,35)(27,36)(33,39)(37,40)7672 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×#2164×C137, C137×#33419, C137×#33420, … |
| maximal normal subgroup of | C2×#2164×C137, C137×#33419, C137×#33420, … |
| maximal quotient of | C137×#12592, C137×#12600, C137×#12654, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17536, 896);