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Cayley diagram

C22⋊(C31⋊C3) #67318

order372
isoclinismisoclinic class of C22⋊(C31⋊C3)
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect productC22⋊(C31⋊C3)
unambiguous semidirect productC31⋊A4

Splittings

Fitting, socle, derived, HallC22×C31 ⋊ C3
SylowC31 ⋊ A4
SylowC22 ⋊ C31⋊C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC22×C31C3
LCSG ▷ C22×C31
UCSC1

10 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 35
transitive degreeμt(G) = 124

A minimal faithful permutation representation — 2 generators on 35 points:

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

Polycyclic presentation:

3,2^2,31|b^a=c,c^a=bc,d^a=d^25

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD31⋊A4, C2×C22⋊(C31⋊C3), A4×C31⋊C3, …
maximal normal subgroup ofD31⋊A4, C2×C22⋊(C31⋊C3), A4×C31⋊C3, …
maximal quotient ofQ8⋊(C31⋊C3), C2×C22⋊(C31⋊C3), C22⋊(C31⋊C9), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(372, 11);