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Cayley diagram

C3×#563 #5893

order192
nilpotency class2
isoclinismisoclinic class of #560
automorphismAut(G) = C2×#72209716 (order 2048) · Inn(G) = C24 · Out(G) = C2×B(4,𝔽2)
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C24

Aliases

direct productC3×#563

Subgroups

subgroupG/H
center, socleC22×C3C24
commutator, Frattini, cosocleC22C24×C3
prime radicalC2×C3×D4C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ G

196 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 4 generators on 27 points:

(1,2)(3,5)(4,6)(7,8)(9,10,13,16)(11,17,19,14)(12,15,21,24)(18,23,20,22)(25,26,27);(1,3)(2,5)(4,7)(6,8)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22);(2,6)(3,7)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23);(2,6)(5,8)(10,16)(11,19)(15,24)(18,20)

Conjugacy classes & character table

57 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#71955, #71956, #71957, …
maximal subgroup of#71955, #71956, #71957, …
maximal normal subgroup of#71955, #71956, #71957, …
maximal quotient ofC3×#2376, C3×#2378, C3×#2379, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1452);