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Cayley diagram

C2×#4430 #67374

order378
isoclinismisoclinic class of #4427
automorphismAut(G) = H(3,𝔽3)×AGL(1,𝔽7) (order 1134) · Inn(G) = C3×ΓL(1,𝔽8) · Out(G) = C2×C32
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C1

Aliases

direct productC2×#4430

Splittings

HallC2×C7 ⋊ 3−1+2
SylowC7 ⋊ C2×3−1+2

Subgroups

subgroupG/H
centerC2×C3C3×ΓL(1,𝔽8)
commutator, cosocleC3×C7C2×C32
FrattiniC3C2×C3×ΓL(1,𝔽8)
Fitting, prime radicalC2×C32×C7C3
socleC2×C3×C7C32
hypercenterC2×C32ΓL(1,𝔽8)
hypocenterC7C2×3−1+2
LCSG ▷ C3×C7 ▷ C7
UCSC1 ◁ C2×C3 ◁ C2×C32

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 126

A minimal faithful permutation representation — 2 generators on 18 points:

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

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#4430⋊C4, C4×#4430, C2×#4430⋊C2, …
maximal normal subgroup of#4430⋊C4, C4×#4430, C2×#4430⋊C2, …
maximal quotient ofC4×#4430, C22×#4430, C2×(C7⋊C9)⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(378, 27);