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C2×C4×C37⋊C3 #11725164

order888
isoclinismisoclinic class of C37⋊C3
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC2×C4×C37⋊C3
unambiguous semidirect product(C2×C4×C37)⋊C3

Splittings

Fitting, HallC2×C4×C37 ⋊ C3
derived, SylowC37 ⋊ C2×C3×C4

Subgroups

subgroupG/H
center, hypercenterC2×C4C37⋊C3
commutator, hypocenterC37C2×C3×C4
FrattiniC2C22×C37⋊C3
FittingC2×C4×C37C3
socleC22×C37C2×C3
prime radicalC22×C37⋊C3C2
cosocleC2×C37C22×C3
LCSG ▷ C37
UCSC1 ◁ C2×C4

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 43
transitive degreeμt(G) = 296

A minimal faithful permutation representation — 2 generators on 43 points:

(1,2)(8,9,11)(10,13,18)(12,16,15)(14,20,27)(17,22,24)(19,25,21)(23,30,34)(26,32,31)(28,35,29)(33,38,39)(36,40,37)(41,43,42);(3,4,5,6)(7,8,10,14,20,28,36,41,40,25,11,15,21,29,30,13,19,26,33,35,18,24,31,37,38,22,9,12,17,23,27,34,39,42,43,32,16)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#64436⋊C3, #10596934⋊C3, C4×Dic37⋊C3, …
maximal normal subgroup ofC2×#64436⋊C3, #10596934⋊C3, C4×Dic37⋊C3, …
maximal quotient ofC42×C37⋊C3, C22⋊C4×C37⋊C3, C4⋊C4×C37⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(888, 22);