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Cayley diagram

C2×C3×M32 #5377

order192
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = C22×#3025 (order 512) · Inn(G) = C22 · Out(G) = C24×D4
homologyAbelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C22

Aliases

direct productC2×C3×M32

Subgroups

subgroupG/H
centerC2×C3×C8C22
commutatorC2C22×C3×C8
Frattini, cosocleC8C23×C3
socleC22×C3C2×C8
prime radicalC23×C3C8
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C8 ◁ G

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 3 generators on 21 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#67809, #67810, #67811, …
maximal subgroup of#67809, #67810, #67811, …
maximal normal subgroup of#67809, #67810, #67811, …
maximal quotient ofC2×C3×#346, C3×C4×M32, C2×C3×C22⋊C16, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 936);