Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| derived | #5464 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #11715754 |
| commutator, hypocenter | #5464 | C2×C4 |
| Frattini, socle | C23 | C2×#1020 |
| Fitting | C2×#3549 | D3 |
| prime radical | C2×#3549⋊C3 | C2 |
| cosocle | C2×#5464 | C22 |
| LCS | G ▷ #5464 | |
| UCS | C1 ◁ C2 | |
1441 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 1441 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2,3,4)(5,9,8,12)(6,17,10,19)(7,28,14,20,16,24,22,27)(11,23,18,25,15,26,21,13);(5,11,13,6,7,20)(8,16,27,10,15,25)(9,14,23,17,18,24)(12,21,28,19,22,26)Polycyclic presentation:
2^3,3,2^6|b^2=c,d^b=d^2,e^a=ej,e^b=ej,e^d=f,f^a=fij,f^b=efi,f^d=ef,g^a=gj,g^d=h,g^e=gj,g^f=gi,h^a=hij,h^b=gh,h^d=gh,h^e=hi,h^f=hij,i^b=ij,i^d=j,j^d=ij48 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#11715754, #72310711, #72310749, … |
| maximal normal subgroup of | C4×#11715754, #72310711, #72310749, … |
| maximal quotient of | C2×#72310668 |
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, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.5*F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 2, F.7*F.8);;
gap> SetConjugate(coll, 8, 4, F.7*F.8);;
gap> SetConjugate(coll, 8, 5, F.8*F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.10);;
gap> SetConjugate(coll, 10, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);