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Cayley diagram

C2×D111 #89233

order444
isoclinismisoclinic class of D111
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD222
direct productC2×D111
unambiguous semidirect product(C2×C3)⋊D37
unambiguous semidirect product(C2×C37)⋊D3

Splittings

Fitting, socleC2×C3×C37 ⋊ C2
derived, HallC3×C37 ⋊ C22
SylowC37 ⋊ C2×D3
SylowC3 ⋊ C2×D37

Subgroups

subgroupG/H
center, hypercenterC2D111
commutator, cosocle, hypocenterC3×C37C22
FrattiniC1G
Fitting, socleC2×C3×C37C2
LCSG ▷ C3×C37
UCSC1 ◁ C2

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 42
transitive degreeμt(G) = 222

A minimal faithful permutation representation — 2 generators on 42 points:

(1,2)(3,4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42);(4,5)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)

Conjugacy classes & character table

114 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11725159, #11725161, #11725162, …
maximal normal subgroup of#11725159, #11725161, #11725162, …
maximal quotient ofDic222, C4×D111, D444, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(444, 17);