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C3×#6398 #10620581

order672
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC3×#6398

Splittings

HallC3×C7 ⋊ #101
SylowC7 ⋊ C3×#101

Subgroups

subgroupG/H
centerC2×C3(C2×D7)⋊C4
commutatorC22×C7C2×C3×C4
FrattiniC2×C4C2×C3×D7
FittingC3×C7×M16C2
socleC2×C3×C7C22⋊C4
prime radicalC22×C3×C7C2×C4
cosocleC2×C4×C7C22×C3
hypercenterC3×M16D7
hypocenterC7C3×#101
LCSG ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×M16

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 336

A minimal faithful permutation representation — 2 generators on 26 points:

(1,2,3)(4,5,7,9,10,8,6)(11,13,14,20,15,21,23,26)(12,16,17,24,18,19,22,25);(5,6)(7,8)(9,10)(11,12,15,18)(13,19,21,16)(14,22,23,17)(20,24,26,25)

Conjugacy classes & character table

123 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13053568, #13053570, #13053572, …
maximal normal subgroup of#13053568, #13053570, #13053572, …
maximal quotient ofC3×#89260, C3×#89264, C3×#89277, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 163);