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Cayley diagram

C4×#1094 #88685

order432
isoclinismisoclinic class of C9⋊A4
automorphismAut(G) = C2×A4⋊#272 (order 1296) · Inn(G) = C3×A4 · Out(G) = C2×C3⋊D3
homologyAbelianization (H1) = C32×C4 · Schur multiplier (H2) = C2

Aliases

direct productC4×#1094

Splittings

SylowC22×C4 ⋊ 3−1+2

Subgroups

subgroupG/H
centerC3×C4C3×A4
commutatorC22×C3C32×C4
FrattiniC2×C3C2×C3×A4
FittingC22×C32×C4C3
socle, cosocleC23×C3C2×C32
prime radicalC23×C32C2×C3
hypercenterC32×C4A4
hypocenterC22C4×3−1+2
LCSG ▷ C22×C3 ▷ C22
UCSC1 ◁ C3×C4 ◁ C32×C4

90 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)(5,8)(6,7)(10,13,14)(12,16,17);(6,7,8)(9,13,16,11,10,17,15,14,12)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#1094⋊C8, C8×#1094, #1094⋊Q8, …
maximal normal subgroup of#1094⋊C8, C8×#1094, #1094⋊Q8, …
maximal quotient ofC4×#6204, C8×#1094, #11721359, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 332);