Finite Group DatabasebetaFinite Group Databasebeta| order | 13696 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C107 · Schur multiplier (H2) = C26 |
| direct product | C2×#453×C107 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C107 | #120 |
| commutator, Frattini, cosocle | C2×C4 | C24×C107 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C107 ◁ C22×C4×C107 ◁ G | |
844 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 844 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 117 |
| transitive degree | μt(G) = 1712 |
A minimal faithful permutation representation — 4 generators on 117 points:
(1,2)(11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117);(4,7)(5,9)(8,10);(3,4)(5,10)(6,8)(7,9);(5,9)(7,10)3424 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C107×#13047, C107×#13049, C107×#13346, … |
| maximal normal subgroup of | C107×#13047, C107×#13049, C107×#13346, … |
| maximal quotient of | C2×C107×#1595, C2×C107×#1596, C2×C107×#1613, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13696, 1746);