Finite Group Databasebeta

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C37×#454 #14557119

order2368
nilpotency class3
isoclinismisoclinic class of #453
automorphism?
homologyAbelianization (H1) = C23×C37 · Schur multiplier (H2) = C23

Aliases

direct productC37×#454

Subgroups

subgroupG/H
center, socleC2×C37#120
commutator, Frattini, cosocleC2×C4C23×C37
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C37 ◁ C2×C4×C37 ◁ G

146 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)(39,43)(41,48)(44,52)(45,49)(46,50)(51,53);(38,39)(40,44)(41,49)(42,45)(43,48)(46,51)(47,53)(50,52);(38,40,42,47)(39,44,45,53)(41,50,48,46)(43,51,49,52)

Conjugacy classes & character table

592 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC37×#1404, C37×#2202, C37×#2203, …
maximal normal subgroup ofC37×#1404, C37×#2202, C37×#2203, …
maximal quotient ofC37×#1597, C37×#1598, C37×#1599, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2368, 135);