Finite Group Databasebeta

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Cayley diagram

C2×Dic24 #4905

order192
isoclinismisoclinic class of D48
automorphismAut(G) order 6144
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C22

Aliases

direct productC2×Dic24

Splittings

SylowC3 ⋊ C2×Q32

Subgroups

subgroupG/H
centerC22D24
commutator, cosocleC3×C8C23
FrattiniC8C22×D3
FittingC2×C3×C16C2
socle, prime radicalC22×C3D8
hypercenterC2×C16D3
hypocenterC3C2×Q32
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16

82 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 37
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 3 generators on 37 points:

(3,4,5)(6,8,17,33,10,19,23,25,11,20,28,37,22,34,9,18)(7,12,21,31,14,26,29,32,15,27,36,35,24,30,13,16);(1,2)(4,5)(6,7,11,15)(8,16,20,32)(9,21,23,36)(10,24,22,14)(12,25,27,18)(13,28,29,17)(19,35,34,31)(26,33,30,37);(1,2)

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#67578, #67584, #67597, …
maximal normal subgroup of#67578, #67584, #67597, …
maximal quotient of#69044, C4×Dic24, #69065, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 464);