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Cayley diagram

C3×#441 #5315

order192
nilpotency class3
isoclinismisoclinic class of AGL(1,ℤ/8)
automorphismAut(G) = C2×#6628842 (order 1024) · Inn(G) = C2×D4 · Out(G) = C2×#120
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22

Aliases

direct productC3×#441

Subgroups

subgroupG/H
center, socle, prime radicalC22×C3C2×D4
commutatorC4C22×C3×C4
Frattini, cosocleC2×C4C23×C3
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ G

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 3 generators on 23 points:

(1,2,4,3)(5,6)(7,13)(8,16)(9,11)(10,14)(12,18)(15,20)(17,19)(21,22,23);(5,7,9,15)(6,10,11,19)(8,17,18,14)(12,20,16,13);(1,3,4,2)(6,11)(10,19)(12,16)(13,20)

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#70917, #79815, #79816, …
maximal subgroup of#70917, C3×#2180, #79815, …
maximal normal subgroup of#70917, C3×#2180, #79815, …
maximal quotient ofC3×#1579, C3×#1587, C3×#1593, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 874);