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Cayley diagram

C2×#829 #5180

order192
isoclinismisoclinic class of #826
automorphismAut(G) = D3×#10576928 (order 3072) · Inn(G) = #213 · Out(G) = C22×#41
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C22

Aliases

direct productC2×#829

Splittings

SylowC3 ⋊ C2×Q32

Subgroups

subgroupG/H
centerC22#213
commutator, cosocleC3×C8C23
FrattiniC8C22×D3
FittingC2×C3×Q16C2
socle, prime radicalC22×C3D8
hypercenterC2×Q16D3
hypocenterC3C2×Q32
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×Q16

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:

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

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#69207, #69209, #69225, …
maximal normal subgroup of#69207, #69209, #69225, …
maximal quotient ofC2×#4489, C2×#4492, #70603, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 739);