Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #67607 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C3 ⋊ Q128 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #4519 |
| commutator, cosocle | C3×C32 | C22 |
| Frattini | C32 | C2×D3 |
| Fitting | C3×Q64 | C2 |
| socle, prime radical | C2×C3 | D32 |
| hypercenter | Q64 | D3 |
| hypocenter | C3 | Q128 |
| LCS | G ▷ C3×C32 ▷ C3×C16 ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ Q64 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 131 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 2 generators on 131 points:
(1,2,3)(4,6,11,23)(5,12,17,46)(7,25,30,78)(8,32,35,95)(9,37,39,99)(10,41,38,22)(13,48,51,112)(14,53,55,120)(15,57,58,122)(16,59,40,45)(18,60,63,24)(19,47,42,27)(20,52,68,26)(21,56,70,33)(28,85,87,131)(29,89,86,77)(31,43,92,105)(34,97,80,94)(36,44,98,107)(49,114,115,128)(50,116,100,111)(54,121,90,119)(61,125,126,84)(62,74,71,88)(64,113,102,130)(65,83,101,110)(66,123,127,72)(67,81,93,117)(69,124,129,73)(75,109,106,79)(76,108,103,82)(91,118,96,104);(2,3)(4,5,11,17)(6,18,23,63)(7,24,30,60)(8,31,35,92)(9,36,39,98)(10,40,38,16)(12,42,46,19)(13,47,51,27)(14,52,55,26)(15,56,58,33)(20,66,68,127)(21,69,70,129)(22,71,41,62)(25,53,78,120)(28,84,87,125)(29,88,86,74)(32,48,95,112)(34,96,80,91)(37,61,99,126)(43,103,105,76)(44,106,107,75)(45,65,59,101)(49,113,115,130)(50,83,100,110)(54,81,90,117)(57,102,122,64)(67,128,93,114)(72,109,123,79)(73,108,124,82)(77,121,89,119)(85,118,131,104)(94,116,97,111)Polycyclic presentation:
2^7,3|a^2=g,b^2=g,c^2=de,d^2=ef,e^2=fg,f^2=g,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,h^a=h^2,c^b=cd,d^b=de,e^b=ef,f^b=fg54 conjugacy classes — view the full character table.
| maximal subgroup of | D3×Q128, C2×#67610, #10716239, … |
| maximal normal subgroup of | D3×Q128, C2×#67610, #10716239, … |
| maximal quotient of | #10683067, #10683068, #10683086, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 186);