Finite Group Databasebeta

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Cayley diagram

C2×C3×#97 #5277

order192
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = C2×#72219167 (order 4096) · Inn(G) = C22 · Out(G) = C2×#449475
homologyAbelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C23

Aliases

direct productC2×C3×#97

Subgroups

subgroupG/H
centerC22×C3×C4C22
commutatorC2C2×C3×C42
Frattini, cosocleC2×C4C23×C3
socle, prime radicalC23×C3C2×C4
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C3×C4 ◁ G

146 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 3 generators on 17 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#67659, #67660, #67661, …
maximal subgroup of#67659, #67660, #67661, …
maximal normal subgroup of#67659, #67660, #67661, …
maximal quotient ofC2×C3×#322, C3×#1451, C3×#1452, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 836);