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Cayley diagram

C23×#157 #64391

order288
isoclinismisoclinic class of #157
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C3

Aliases

direct productC23×#157

Splittings

Fitting, socleC23×C32 ⋊ C4
derived, SylowC32 ⋊ C23×C4

Subgroups

subgroupG/H
center, hypercenterC23#157
commutator, hypocenterC32C23×C4
FrattiniC1G
Fitting, socleC23×C32C4
prime radicalC23×C3⋊D3C2
cosocleC3⋊D3C24
LCSG ▷ C32
UCSC1 ◁ C23

370 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 4 generators on 12 points:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72252889, #72252941, #72252946, …
maximal subgroup ofC2×#63737, #10593515, #10593516, …
maximal normal subgroup ofC2×#63737, #10593515, #10593516, …
maximal quotient ofC22×#3787, C22×#3788, C2×#64283, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 1039);