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D456 #11744882

order912
automorphismAut(G) order 65664
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD456

Splittings

FittingC3×C8×C19 ⋊ C2
HallC3×C19 ⋊ D8
SylowC19 ⋊ D24
SylowC3 ⋊ D152

Subgroups

subgroupG/H
centerC2D228
commutator, cosocleC3×C4×C19C22
FrattiniC4C2×D57
FittingC3×C8×C19C2
socleC2×C3×C19D4
hypercenterC8D57
hypocenterC3×C19D8
LCSG ▷ C3×C4×C19 ▷ C2×C3×C19 ▷ C3×C19
UCSC1 ◁ C2 ◁ C4 ◁ C8

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 456

A minimal faithful permutation representation — 2 generators on 30 points:

(1,2,3,4,5,6,7,8)(9,10,11)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30);(2,8)(3,7)(4,6)(10,11)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)

Polycyclic presentation:

2^4,3,19|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^18

Conjugacy classes & character table

231 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#14268828, #14268829, #14268833, …
maximal normal subgroup of#14268828, #14268829, #14268833, …
maximal quotient of#14268993, D912, Dic456, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(912, 93);