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Cayley diagram

C2×Dic23 #4405

order184
isoclinismisoclinic class of D23
automorphismAut(G) = D4×AGL(1,𝔽23) (order 4048) · Inn(G) = D23 · Out(G) = D4×C11
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2

Aliases

direct productC2×Dic23
unambiguous semidirect product(C2×C23)⋊C4

Splittings

derived, SylowC23 ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC22D23
commutator, hypocenterC23C2×C4
FrattiniC2C2×D23
Fitting, socle, prime radicalC22×C23C2
cosocleC2×C23C22
LCSG ▷ C23
UCSC1 ◁ C22

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 29 points:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×Dic23, Dic23⋊C4, C4⋊Dic23, …
maximal normal subgroup ofC4×Dic23, Dic23⋊C4, C4⋊Dic23, …
maximal quotient ofC2×C23⋊C8, #67267, C4×Dic23, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(184, 6);