Finite Group Databasebeta

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Cayley diagram

C4⋊C4×C3⋊D3 #64100

order288
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C3

Aliases

direct productC4⋊C4×C3⋊D3

Splittings

FittingC32×C4⋊C4 ⋊ C2
SylowC32 ⋊ C2×C4⋊C4

Subgroups

subgroupG/H
center, FrattiniC22C22×C3⋊D3
commutatorC2×C32C22×C4
FittingC32×C4⋊C4C2
socle, cosocleC22×C32C23
prime radicalC22×C3⋊D3C22
hypercenterC4⋊C4C3⋊D3
hypocenterC32C2×C4⋊C4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

276 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 144

A minimal faithful permutation representation — 3 generators on 14 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10590163, #10590510, #10590511, …
maximal normal subgroup of#10590163, #10590510, #10590511, …
maximal quotient of#10592346, #10592348, #10592349, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 748);