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C2×#66524 #10620390

order666
isoclinismisoclinic class of C37⋊C3
automorphism?
homologyAbelianization (H1) = C2×C9 · Schur multiplier (H2) = C1

Aliases

direct productC2×#66524

Splittings

derived, SylowC37 ⋊ C2×C9
HallC2×C37 ⋊ C9

Subgroups

subgroupG/H
center, hypercenterC2×C3C37⋊C3
commutator, hypocenterC37C2×C9
FrattiniC3C2×C37⋊C3
Fitting, socle, prime radicalC2×C3×C37C3
cosocleC3×C37C2×C3
LCSG ▷ C37
UCSC1 ◁ C2×C3

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 48
transitive degreeμt(G) = 666

A minimal faithful permutation representation — 2 generators on 48 points:

(1,2)(4,5,7)(6,9,14)(8,12,11)(10,16,23)(13,18,20)(15,21,17)(19,26,30)(22,28,27)(24,31,25)(29,34,35)(32,36,33)(37,39,38)(40,41,43,42,44,46,45,47,48);(3,4,6,10,16,24,32,37,36,21,7,11,17,25,26,9,15,22,29,31,14,20,27,33,34,18,5,8,13,19,23,30,35,38,39,28,12)

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13050607, C4×#66524, C2×#66524⋊C2, …
maximal normal subgroup of#13050607, C4×#66524, C2×#66524⋊C2, …
maximal quotient ofC4×#66524, C22×#66524, C2×#11758601, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(666, 2);