Finite Group DatabasebetaFinite Group Databasebeta| order | 352 |
| isoclinism | isoclinic class of #4314 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C8×C11 ⋊ C2 |
| Sylow | C11 ⋊ #135 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | D44 |
| commutator, cosocle | C4×C11 | C23 |
| Frattini | C4 | C22×D11 |
| Fitting | C2×C8×C11 | C2 |
| socle | C2×C11 | C2×D4 |
| hypercenter | C2×C8 | D11 |
| hypocenter | C11 | #135 |
| LCS | G ▷ C4×C11 ▷ C2×C11 ▷ C11 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×C8 | |
62 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 176 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,4,6,8,10,11,9,7,5,3)(12,15,16,23,17,24,14,21)(13,19,18,22,20,26,25,27);(12,14,17,16)(13,18,20,25)(15,21,24,23)(19,22,26,27);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,18)(15,22)(16,25)(17,20)(19,23)(21,26)(24,27)Polycyclic presentation:
2^5,11|b^2=e,c^2=d,d^2=e,c^a=cd,d^a=de,f^a=f^1094 conjugacy classes — view the full character table.
| maximal subgroup of | #10622126, #10622135, #10622148, … |
| maximal normal subgroup of | #10622126, #10622135, #10622148, … |
| maximal quotient of | #10622296, C4×#4314, C4×D88, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(352, 99);