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C7×#805 #10620603

order672
isoclinismisoclinic class of #204
automorphism?
homologyAbelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C2

Aliases

direct productC7×#805

Splittings

FittingC3×C42×C7 ⋊ C2
HallC3×C7 ⋊ #104
SylowC3 ⋊ C7×#104

Subgroups

subgroupG/H
centerC4×C7D12
commutatorC3×C4C2×C4×C7
FrattiniC2×C4C2×D3×C7
FittingC3×C42×C7C2
socleC2×C3×C7C22⋊C4
prime radicalC7×#235C2
cosocleC2×C3×C4C22×C7
hypercenterC42×C7D3
hypocenterC3C7×#104
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4×C7 ◁ C2×C4×C7 ◁ C42×C7

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 168

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

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

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13052481, #13052482, #13052485, …
maximal normal subgroup of#13052481, #13052482, #13052485, …
maximal quotient ofC7×#4451, C7×#4452, C7×#4459, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 185);