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Cayley diagram

#3866⋊C3 #89223

order444
isoclinismisoclinic class of #3866⋊C3
automorphismAut(G) = AGL(1,𝔽37) (order 1332) · Inn(G) = G · Out(G) = C3
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect product#3866⋊C3

Splittings

Fitting, socle, derived, SylowC37 ⋊ C3×C4
Hall#3866 ⋊ C3
HallC37⋊C3 ⋊ C4

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC37C3×C4
prime radicalD37⋊C3C2
cosocleD37C2×C3
LCSG ▷ C37
UCSC1

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 37
transitive degreeμt(G) = 37

A minimal faithful permutation representation — 2 generators on 37 points:

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

Polycyclic presentation:

2,3,2,37|a^2=c,d^a=d^31,d^b=d^26,d^c=d^36

Conjugacy classes & character table

15 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#3866⋊C3, AGL(1,𝔽37), C3×#3866⋊C3, …
maximal normal subgroup ofC2×#3866⋊C3, AGL(1,𝔽37), C3×#3866⋊C3, …
maximal quotient of#64438⋊C3, C2×#3866⋊C3, #13050613, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(444, 7);