Finite Group Databasebeta

← Prev (#6362) Next (#6364) →

Cayley diagram

C2×C37⋊C3 #6363

order222
isoclinismisoclinic class of C37⋊C3
automorphismAut(G) = AGL(1,𝔽37) (order 1332) · Inn(G) = C37⋊C3 · Out(G) = C3×C4
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

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

Splittings

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

Subgroups

subgroupG/H
center, hypercenterC2C37⋊C3
commutator, cosocle, hypocenterC37C2×C3
FrattiniC1G
Fitting, socleC2×C37C3
LCSG ▷ C37
UCSC1 ◁ C2

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 39
transitive degreeμt(G) = 74

A minimal faithful permutation representation — 2 generators on 39 points:

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

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic37⋊C3, C4×C37⋊C3, C2×D37⋊C3, …
maximal normal subgroup ofDic37⋊C3, C4×C37⋊C3, C2×D37⋊C3, …
maximal quotient ofC4×C37⋊C3, C22×C37⋊C3, C2×#66524, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(222, 2);