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C37×#494 #14557159

order2368
nilpotency class3
isoclinismisoclinic class of #465
automorphism?
homologyAbelianization (H1) = C23×C37 · Schur multiplier (H2) = C2×C4

Aliases

direct productC37×#494

Subgroups

subgroupG/H
center, socle, prime radicalC22×C37C2×D4
commutator, Frattini, cosocleC2×C4C23×C37
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C37 ◁ C42×C37 ◁ G

122 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 57
transitive degreeμt(G) = 2368

A minimal faithful permutation representation — 3 generators on 57 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)(39,41)(42,43,46,49)(44,50,53,57)(45,54,51,48)(47,55,56,52);(42,44,51,55,46,53,45,52)(43,47,54,57,49,56,48,50);(38,39,40,41)

Conjugacy classes & character table

814 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC37×#1358, C37×#1362, C37×#1411, …
maximal normal subgroup ofC37×#1358, C37×#1362, C37×#1411, …
maximal quotient ofC37×#1707, C37×#1710, C37×#1713, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2368, 175);