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Cayley diagram

C42×3−1+2 #88555

order432
nilpotency class2
isoclinismisoclinic class of H(3,𝔽3)
automorphism?
homologyAbelianization (H1) = C32×C42 · Schur multiplier (H2) = C4

Aliases

direct productC42×3−1+2

Subgroups

subgroupG/H
centerC3×C42C32
commutatorC3C32×C42
Frattini, socle, cosocleC22×C3C22×C32
prime radicalC22×C32C22×C3
LCSG ▷ C3 ▷ C1
UCSC1 ◁ C3×C42 ◁ G

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 144

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

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC4×3−1+2⋊C8, #63362⋊C3, C4⋊(3−1+2⋊C8), …
maximal subgroup ofC4×3−1+2⋊C8, #63362⋊C3, C4⋊(3−1+2⋊C8), …
maximal normal subgroup ofC4×3−1+2⋊C8, #63362⋊C3, C4⋊(3−1+2⋊C8), …
maximal quotient of3−1+2×#95, C4×C8×3−1+2, 3−1+2×#97, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 202);