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Cayley diagram

C3×M64 #4617

order192
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = C22×#131 (order 128) · Inn(G) = C22 · Out(G) = C23×C4
homologyAbelianization (H1) = C2×C3×C16 · Schur multiplier (H2) = C1

Aliases

direct productC3×M64

Subgroups

subgroupG/H
centerC3×C16C22
commutatorC2C2×C3×C16
Frattini, cosocleC16C22×C3
socleC2×C3C2×C16
prime radicalC22×C3C16
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C3×C16 ◁ G

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 35
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 2 generators on 35 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#67599, #67600, #67601, …
maximal subgroup of#67599, #67600, #67601, …
maximal normal subgroup of#67599, #67600, #67601, …
maximal quotient ofC3×#1397, C3×C22⋊C32, C3×C4⋊C32, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 176);