Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24×C3 |
No aliases for this group.
| Fitting | H(7,𝔽2) ⋊ C3⋊D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2 | C22×A4⋊S4 |
| commutator, cosocle, hypocenter | Ω+(4,𝔽3) | C23 |
| Fitting | H(7,𝔽2) | C3⋊D3 |
| hypercenter | D4 | A4⋊S4 |
| LCS | G ▷ Ω+(4,𝔽3) | |
| UCS | C1 ◁ C2 ◁ D4 | |
1842 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 1842 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,3,38,20,34,31,7,16,30,5,19,36)(2,21,48,11,42,47,8,9,44,23,28,45)(4,37,32,18,14,33,17,15,29,6,35,13)(10,27,46,12,26,25,22,41,43,24,40,39);(3,13)(4,17)(5,18)(6,20)(9,25)(10,22)(11,24)(12,23)(14,32)(15,31)(16,33)(19,38)(21,39)(26,46)(27,45)(28,48)(29,35)(30,34)(36,37)(40,43)(41,47)(42,44);(1,2)(3,9)(4,11,6,10,5,12)(7,8)(13,25)(14,27,19,26,15,28)(16,21)(17,23,18,22,20,24)(29,44,36,43,30,47)(31,46,38,45,32,48)(33,39)(34,40,37,42,35,41)Polycyclic presentation:
2^3,3^2,2^5|f^2=j,g^2=j,c^a=cj,d^b=d^2,e^b=e^2,f^b=fj,f^d=g,f^e=g,g^b=fg,g^d=fgj,g^e=fgj,g^f=gj,h^b=ghij,h^d=i,h^e=fgh,h^g=hj,i^b=fij,i^d=hi,i^e=fij,i^f=ij62 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.7);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 2, F.6*F.7);;
gap> SetConjugate(coll, 7, 4, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.9);;
gap> SetConjugate(coll, 8, 5, F.6*F.7*F.8);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.8*F.9);;
gap> SetConjugate(coll, 9, 5, F.6*F.9*F.10);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);