Finite Group Databasebeta

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Cayley diagram

C22×C4⋊Dic3 #5785

order192
isoclinismisoclinic class of Dic6
automorphismAut(G) order 294912
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C26

Aliases

direct productC22×C4⋊Dic3

Splittings

SylowC3 ⋊ C22×C4⋊C4

Subgroups

subgroupG/H
centerC24C2×D3
commutatorC2×C3C23×C4
FrattiniC22C23×D3
FittingC23×C3×C4C2
socle, prime radicalC24×C3C22
cosocleC22×C3C24
hypercenterC23×C4D3
hypocenterC3C22×C4⋊C4
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C24 ◁ C23×C4

418 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 4 generators on 15 points:

(5,6,7)(12,13,14,15);(6,7)(8,9,10,11)(13,15);(3,4);(1,2)

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#69407, #69410, #69411, …
maximal normal subgroup of#69407, #69410, #69411, …
maximal quotient ofC22×C4⋊(C3⋊C8), C2×#4924, C2×C4×C4⋊Dic3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1344);