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C37×#509 #14557174

order2368
nilpotency class4
isoclinismisoclinic class of #509
automorphism?
homologyAbelianization (H1) = C23×C37 · Schur multiplier (H2) = C22

Aliases

direct productC37×#509

Subgroups

subgroupG/H
center, socleC2×C37C2×D8
commutator, Frattini, cosocleC8C23×C37
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C37 ◁ C2×C4×C37 ◁ C2×C8×C37 ◁ G

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 53
transitive degreeμt(G) = 592

A minimal faithful permutation representation — 3 generators on 53 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37)(38,39)(40,43)(41,50)(42,44)(45,47)(46,48)(49,51)(52,53);(39,43)(40,47)(41,48)(44,51)(45,46)(49,53)(50,52);(39,44)(43,51)(45,52)(46,50)

Conjugacy classes & character table

592 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC37×AGL(1,ℤ/16), C37×#2190, C37×#2192, …
maximal normal subgroup ofC37×AGL(1,ℤ/16), C37×#2190, C37×#2192, …
maximal quotient ofC37×#2139, C37×#2140, C37×#2159, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2368, 190);