Finite Group Databasebeta

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Cayley diagram

C2×C3×#108 #5303

order192
nilpotency class3
isoclinismisoclinic class of D8
automorphismAut(G) = C22×#23882 (order 1024) · Inn(G) = D4 · Out(G) = C22×#120
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22

Aliases

direct productC2×C3×#108

Subgroups

subgroupG/H
centerC2×C3×C4D4
commutatorC4C22×C3×C4
Frattini, cosocleC2×C4C23×C3
socleC22×C3C4⋊C4
prime radicalC23×C3Q8
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4 ◁ G

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:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#67728, #67729, #67730, …
maximal subgroup of#67728, #67729, #67730, …
maximal normal subgroup of#67728, #67729, #67730, …
maximal quotient ofC2×C3×#334, C2×C3×#335, C3×#1564, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 862);