Finite Group DatabasebetaFinite Group Databasebeta| order | 1944 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(5,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C34 · Schur multiplier (H2) = C2×C35 |
| direct product | D4×3−1+4 |
| unambiguous semidirect product | C4⋊(C2×3−1+4) |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C2×C3 | C22×C34 |
| prime radical | C3×D4×H(3,𝔽3) | C3 |
| LCS | G ▷ C2×C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ G | |
1840 conjugacy classes of subgroups; only the 1278 normal subgroups are shown.
This group has 1840 conjugacy classes of subgroups — too many to draw. Only its 1278 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3,4)(5,6,9,8,12,19,17,11,23)(7,13,24,16,22,18,15,10,20)(14,26,30,27,21,31,28,25,29);(2,4)(5,7,14)(6,10,21)(8,16,27)(9,18,29)(11,22,26)(12,13,25)(15,28,17)(19,20,30)(23,24,31);(7,15,16)(10,22,13)(14,27,28)(18,24,20)(21,25,26)(29,30,31);(6,11,12)(9,19,23)(10,22,13)(18,20,24)(21,26,25)(29,30,31)415 conjugacy classes — view the full character table.
| Fitting subgroup of | D4×#4207⋊C3, D4×#92394, D4×(#4207⋊C2)⋊C3 |
| maximal subgroup of | D4×#10624749, D4×#4207⋊C3, D4×#92394, … |
| maximal normal subgroup of | D4×#10624749, D4×#4207⋊C3, D4×#92394 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1944, 3427);