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Cayley diagram

C4×C22⋊C9 #3691

order144
isoclinismisoclinic class of A4
automorphismAut(G) = C2×C3×S4 (order 144) · Inn(G) = A4 · Out(G) = C22×C3
homologyAbelianization (H1) = C4×C9 · Schur multiplier (H2) = C2

Aliases

direct productC4×C22⋊C9
unambiguous semidirect product(C22×C4)⋊C9

Splittings

derivedC22 ⋊ C4×C9
SylowC22×C4 ⋊ C9

Subgroups

subgroupG/H
center, hypercenterC3×C4A4
commutator, hypocenterC22C4×C9
FrattiniC2×C3C2×A4
FittingC22×C3×C4C3
socle, prime radical, cosocleC23×C3C2×C3
LCSG ▷ C22
UCSC1 ◁ C3×C4

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 36

A minimal faithful permutation representation — 2 generators on 17 points:

(1,2,3,4)(6,7,8)(9,16,15,11,10,17,14,13,12);(5,8)(6,7)

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

derived subgroup of#10589500, #10589503, #10589504, …
maximal subgroup of(C22⋊C9)⋊C8, C8×C22⋊C9, (C22⋊C9)⋊Q8, …
maximal normal subgroup of(C22⋊C9)⋊C8, C8×C22⋊C9, (C22⋊C9)⋊Q8, …
maximal quotient ofC4×Q8⋊C9, C8×C22⋊C9, #131⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 34);