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C37×#354 #14557019

order2368
nilpotency class4
isoclinismisoclinic class of #353
automorphism?
homologyAbelianization (H1) = C2×C4×C37 · Schur multiplier (H2) = C22

Aliases

direct productC37×#354

Subgroups

subgroupG/H
center, socleC2×C37#99
commutatorC2×C4C2×C4×C37
Frattini, cosocleC2×D4C22×C37
prime radicalC2×D4×C37C22
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C37 ◁ C22×C37 ◁ C2×D4×C37 ◁ G

70 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 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)(40,46,48,53)(41,49,51,44)(42,45)(43,47)(50,52);(38,40,42,48)(39,43,45,52)(41,50,51,47)(44,53,49,46)

Conjugacy classes & character table

481 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC37×#1404, C37×#1406, C37×#1407, …
maximal normal subgroup ofC37×#1404, C37×#1406, C37×#1407, …
maximal quotient ofC37×#1319, C37×#1324, C37×#1344, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2368, 35);