Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | Aut(G) = S4×AGL(1,ℤ/16) (order 3072) · Inn(G) = C22×D8 · Out(G) = C2×C4×D3 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×D8 |
| commutator, Frattini, cosocle | C8 | C24 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ #131 ◁ G | |
| LECS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
175 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12)(3,13,17,41)(4,9,18,32)(5,21,25,39)(6,26,23,11)(8,28,31,15)(10,35,37,24)(14,40,42,62)(16,45,27,36)(19,48,46,61)(20,51,47,34)(22,38,53,30)(29,54,55,43)(33,58,57,50)(44,63,52,59)(49,60,64,56);(1,3,7,17)(2,8,12,31)(4,14,18,42)(5,22,25,53)(6,27,23,16)(9,29,32,55)(10,36,37,45)(11,38,26,30)(13,39,41,21)(15,35,28,24)(19,49,46,64)(20,52,47,44)(33,59,57,63)(34,60,51,56)(40,61,62,48)(43,58,54,50);(1,4)(2,9)(3,14)(5,19)(6,20)(7,18)(8,29)(10,33)(11,34)(12,32)(13,40)(15,43)(16,44)(17,42)(21,48)(22,49)(23,47)(24,50)(25,46)(26,51)(27,52)(28,54)(30,56)(31,55)(35,58)(36,59)(37,57)(38,60)(39,61)(41,62)(45,63)(53,64);(4,18)(9,32)(14,42)(19,46)(20,47)(29,55)(33,57)(34,51)(40,62)(43,54)(44,52)(48,61)(49,64)(50,58)(56,60)(59,63)Polycyclic presentation:
2^7|a^2=g,b^2=g,e^2=fg,f^2=g,b^a=be,e^a=ef,f^a=fg,e^b=ef,f^b=fg,d^c=dg32 conjugacy classes — view the full character table.
| Fitting subgroup of | #3417⋊C3, #11711667, #11711669, … |
| maximal subgroup of | #13670, #13672, #33855, … |
| maximal normal subgroup of | #13670, #13672, #33855, … |
| maximal quotient of | #33382, #33410, #33455, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2149);