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Cayley diagram

C2×C22⋊D23 #67294

order368
isoclinismisoclinic class of Dic46
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23

Aliases

direct productC2×C22⋊D23
unambiguous semidirect productC23⋊D23

Splittings

FittingC23×C23 ⋊ C2
SylowC23 ⋊ C2×D4

Subgroups

subgroupG/H
centerC22C2×D23
commutator, cosocleC2×C23C23
FrattiniC2C22×D23
FittingC23×C23C2
socleC22×C23C22
hypercenterC23D23
hypocenterC23C2×D4
LCSG ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C23

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 29
transitive degreeμt(G) = 184

A minimal faithful permutation representation — 3 generators on 29 points:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10624988, (C22⋊D23)⋊C4, #10625053, …
maximal normal subgroup of#10624988, (C22⋊D23)⋊C4, #10625053, …
maximal quotient ofC2×Dic23⋊C4, #10625095, C2×(C2×D23)⋊C4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(368, 36);