Finite Group Databasebeta

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Cayley diagram

C2×#822 #5123

order192
isoclinismisoclinic class of #213
automorphismAut(G) = C2×D3×#3484 (order 1536) · Inn(G) = #67 · Out(G) = C23×D4
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C22

Aliases

direct productC2×#822

Splittings

SylowC3 ⋊ C2×#108

Subgroups

subgroupG/H
centerC2×C4#67
commutatorC3×C4C22×C4
FrattiniC2×C4C22×D3
FittingC2×C3×M16C2
socleC22×C3C4⋊C4
prime radicalC23×C3Q8
cosocleC2×C3×C4C23
hypercenterC2×M16D3
hypocenterC3C2×#108
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×M16

106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 3 generators on 21 points:

(3,4,5)(6,8,9,17,11,18,10,19)(7,12,13,16,15,20,14,21);(4,5)(6,7,10,14,11,15,9,13)(8,16,17,20,18,21,19,12);(1,2)

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#68699, #68724, #68741, …
maximal normal subgroup of#68699, #68724, #68741, …
maximal quotient ofC4×#822, #69875, #69982, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 682);