Finite Group Databasebeta

← Prev (#13050323) Next (#13050325) →

Dic6×C11⋊C5 #13050324

order1320
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C1

Aliases

direct productDic6×C11⋊C5
unambiguous semidirect productC3⋊(Q8×C11⋊C5)
unambiguous semidirect product(C11×Dic6)⋊C5

Splittings

HallC11×Dic6 ⋊ C5
HallC3×C11⋊C5 ⋊ Q8
HallC3×C11 ⋊ C5×Q8
SylowC11 ⋊ C5×Dic6
SylowC3 ⋊ Q8×C11⋊C5

Subgroups

subgroupG/H
center, FrattiniC2C2×D3×C11⋊C5
commutator, socle, cosocleC2×C3×C11C22×C5
FittingC3×C4×C11C2×C5
prime radicalC2×C3×C11⋊C5C22
hypercenterC4D3×C11⋊C5
hypocenterC3×C11C5×Q8
LCSG ▷ C2×C3×C11 ▷ C3×C11
UCSC1 ◁ C2 ◁ C4

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 264

A minimal faithful permutation representation — 2 generators on 22 points:

(1,2,3,4)(5,8,7,6)(9,10,11)(13,14,16,19,18)(15,17,21,22,20);(1,5,3,7)(2,6,4,8)(10,11)(12,13,15,18,14,16,20,22,17,19,21)

Conjugacy classes & character table

63 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#204×C11⋊C5, #72216808, #214×C11⋊C5, …
maximal normal subgroup of#204×C11⋊C5, #72216808, #214×C11⋊C5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1320, 28);