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Cayley diagram

C22⋊(C37⋊C3) #89230

order444
isoclinismisoclinic class of C22⋊(C37⋊C3)
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect productC22⋊(C37⋊C3)
unambiguous semidirect productC37⋊A4

Splittings

Fitting, socle, derived, HallC22×C37 ⋊ C3
SylowC37 ⋊ A4
SylowC22 ⋊ C37⋊C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC22×C37C3
LCSG ▷ C22×C37
UCSC1

10 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 41
transitive degreeμt(G) = 148

A minimal faithful permutation representation — 2 generators on 41 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD37⋊A4, C2×C22⋊(C37⋊C3), A4×C37⋊C3, …
maximal normal subgroup ofD37⋊A4, C2×C22⋊(C37⋊C3), A4×C37⋊C3, …
maximal quotient ofQ8⋊(C37⋊C3), C2×C22⋊(C37⋊C3), C22⋊#66524, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(444, 14);