Finite Group Databasebeta

← Prev (#5380) Next (#5382) →

Cayley diagram

C2×C3×Q32 #5381

order192
nilpotency class4
isoclinismisoclinic class of D16
automorphismAut(G) = C2×#72213937 (order 2048) · Inn(G) = D8 · Out(G) = C2×#525
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C22

Aliases

direct productC2×C3×Q32

Subgroups

subgroupG/H
center, socle, prime radicalC22×C3D8
commutator, Frattini, cosocleC8C23×C3
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8 ◁ G

82 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 37
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 3 generators on 37 points:

(1,2)(35,36,37);(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)(19,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20);(3,19,11,27)(4,20,12,28)(5,21,13,29)(6,22,14,30)(7,23,15,31)(8,24,16,32)(9,25,17,33)(10,26,18,34)

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#67826, #67827, C2×#4521, …
maximal subgroup of#67826, #67827, C3×#1416, …
maximal normal subgroup of#67826, #67827, C3×#1416, …
maximal quotient ofC2×C3×#358, C2×C3×#366, C3×C4×Q32, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 940);