Finite Group DatabasebetaFinite Group Databasebeta| order | 392 |
| isoclinism | isoclinic class of #87656 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C7 |
No aliases for this group.
| Fitting, socle, derived, Sylow | C72 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C72 | C8 |
| prime radical | C7⋊D7 | C4 |
| cosocle | #5997 | C2 |
| LCS | G ▷ C72 | |
| UCS | C1 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 28 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2,3,5)(4,7,9,13,10,6,14,15)(8,12,18,23,19,24,22,28)(11,17,20,25,21,16,26,27);(1,4,11,22,18,21,10)(2,28,6,17,16,7,23)(3,9,20,8,19,26,14)Polycyclic presentation:
2^3,7^2|a^2=b,b^2=c,d^a=d^4e^5,e^a=d^6e^6,d^b=d^4e,e^b=d^4e^3,d^c=d^6,e^c=e^614 conjugacy classes — view the full character table.
| derived subgroup of | #11716140⋊C2 |
| maximal subgroup of | #11716140, #87656⋊C2, #11716142, … |
| maximal normal subgroup of | #11716140, #87656⋊C2, #11716142, … |
| maximal quotient of | #11716084, C2×#87656, C3×#87656, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(392, 36);