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Cayley diagram

C4×C22⋊C27 #88393

order432
isoclinismisoclinic class of A4
automorphismAut(G) = C2×C9×S4 (order 432) · Inn(G) = A4 · Out(G) = C22×C9
homologyAbelianization (H1) = C4×C27 · Schur multiplier (H2) = C2

Aliases

direct productC4×C22⋊C27
unambiguous semidirect product(C22×C4)⋊C27

Splittings

derivedC22 ⋊ C4×C27
SylowC22×C4 ⋊ C27

Subgroups

subgroupG/H
center, hypercenterC4×C9A4
commutator, hypocenterC22C4×C27
FrattiniC2×C9C2×A4
FittingC22×C4×C9C3
socle, prime radicalC23×C3C2×C9
cosocleC23×C9C2×C3
LCSG ▷ C22
UCSC1 ◁ C4×C9

45 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 35
transitive degreeμt(G) = 108

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

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

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

derived subgroup of#13136136, #13136139, #13136140, …
maximal subgroup of(C22⋊C27)⋊C8, C8×C22⋊C27, (C22⋊C27)⋊Q8, …
maximal normal subgroup of(C22⋊C27)⋊C8, C8×C22⋊C27, (C22⋊C27)⋊Q8, …
maximal quotient ofC4×Q8⋊C27, C8×C22⋊C27, #131⋊C27, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 40);