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C2×D253 #11759804

order1012
isoclinismisoclinic class of D253
automorphismAut(G) order 111320
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD506
direct productC2×D253
unambiguous semidirect product(C2×C11)⋊D23
unambiguous semidirect product(C2×C23)⋊D11

Splittings

Fitting, socleC2×C11×C23 ⋊ C2
derived, HallC11×C23 ⋊ C22
SylowC23 ⋊ C2×D11
SylowC11 ⋊ C2×D23

Subgroups

subgroupG/H
center, hypercenterC2D253
commutator, cosocle, hypocenterC11×C23C22
FrattiniC1G
Fitting, socleC2×C11×C23C2
LCSG ▷ C11×C23
UCSC1 ◁ C2

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 36
transitive degreeμt(G) = 506

A minimal faithful permutation representation — 2 generators on 36 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);(4,13)(5,12)(6,11)(7,10)(8,9)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×D253, C2×D759, C2×C3×D253, …
maximal normal subgroup ofC2×C3×D253, C2×C11⋊AGL(1,𝔽23), C2×C11×D253, …
maximal quotient ofD3×D253, C2×D759, C2×C3×D253, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1012, 12);