Finite Group Databasebeta

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Cayley diagram

C3×Dic38 #90704

order456
isoclinismisoclinic class of Dic38
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C1

Aliases

direct productC3×Dic38

Splittings

HallC3×C19 ⋊ Q8
SylowC19 ⋊ C3×Q8

Subgroups

subgroupG/H
centerC2×C3C2×D19
commutator, cosocleC2×C19C22×C3
FrattiniC2C2×C3×D19
FittingC3×C4×C19C2
socle, prime radicalC2×C3×C19C22
hypercenterC3×C4D19
hypocenterC19C3×Q8
LCSG ▷ C2×C19 ▷ C19
UCSC1 ◁ C2×C3 ◁ C3×C4

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 456

A minimal faithful permutation representation — 2 generators on 30 points:

(1,2,3)(4,5,6,7)(8,11,10,9)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30);(4,8,6,10)(5,9,7,11)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)

Conjugacy classes & character table

123 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11744822, #11744824, #11744827, …
maximal normal subgroup of#11744822, #11744824, #11744827, …
maximal quotient ofC3×Dic19⋊C4, C3×C4⋊Dic19, C2×C3×Dic38, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(456, 24);