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Cayley diagram

C3×C4×M16 #5278

order192
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = C2×#72219157 (order 2048) · Inn(G) = C22 · Out(G) = C2×#35680
homologyAbelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C2×C4

Aliases

direct productC3×C4×M16

Subgroups

subgroupG/H
centerC3×C42C22
commutatorC2C2×C3×C42
Frattini, cosocleC2×C4C23×C3
socleC22×C3C22×C4
prime radicalC23×C3C2×C4
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C3×C42 ◁ G

142 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 3 generators on 15 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#67665, #67666, #67667, …
maximal subgroup of#67665, #67666, #67667, …
maximal normal subgroup of#67665, #67666, #67667, …
maximal quotient ofC3×C8×M16, C3×#1451, C3×#1453, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 837);