Finite Group Databasebeta

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Cayley diagram

C22×C3⋊Dic3 #3833

order144
isoclinismisoclinic class of C3⋊D3
automorphismAut(G) order 82944
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C3

Aliases

direct productC22×C3⋊Dic3

Splittings

derived, SylowC32 ⋊ C22×C4

Subgroups

subgroupG/H
center, hypercenterC23C3⋊D3
commutator, hypocenterC32C22×C4
FrattiniC2C22×C3⋊D3
Fitting, socle, prime radicalC23×C32C2
cosocleC2×C32C23
LCSG ▷ C32
UCSC1 ◁ C23

162 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:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

derived subgroup of#11771793, #11771794, #11771810, …
maximal subgroup of#63579, #63658, C22⋊#622, …
maximal normal subgroup of#63579, #63658, C22⋊#622, …
maximal quotient ofC22×C3⋊(C3⋊C8), C2×#3748, C2×C4×C3⋊Dic3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 176);