Finite Group Databasebeta

← Prev (#5248) Next (#5250) →

Cayley diagram

C2×C3×#95 #5249

order192
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) order 24576
homologyAbelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C25

Aliases

direct productC2×C3×#95

Subgroups

subgroupG/H
center, socle, prime radicalC24×C3C22
commutatorC2C2×C3×C42
Frattini, cosocleC23C23×C3
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

330 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);(11,13)(14,15,16,17);(6,7,8,9)(10,11)(12,13)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#67612, #67613, C2×#4451, …
maximal subgroup of#67612, #67613, C3×#1280, …
maximal normal subgroup of#67612, #67613, C3×#1280, …
maximal quotient ofC3×C4×#95, C3×C22⋊#95, C3×C4⋊#95, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 808);