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Cayley diagram

C22×C31⋊C3 #67316

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

Aliases

direct productC22×C31⋊C3

Splittings

Fitting, socle, HallC22×C31 ⋊ C3
derived, SylowC31 ⋊ C22×C3

Subgroups

subgroupG/H
center, hypercenterC22C31⋊C3
commutator, cosocle, hypocenterC31C22×C3
FrattiniC1G
Fitting, socleC22×C31C3
LCSG ▷ C31
UCSC1 ◁ C22

20 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:

(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);(1,2)(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)

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×(C31⋊C3)⋊C4, (C22⋊D31)⋊C3, C2×C4×C31⋊C3, …
maximal normal subgroup ofC2×(C31⋊C3)⋊C4, (C22⋊D31)⋊C3, C2×C4×C31⋊C3, …
maximal quotient ofC2×C4×C31⋊C3, D4×C31⋊C3, Q8×C31⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(372, 9);