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C41×#501 #14579969

order2624
nilpotency class3
isoclinismisoclinic class of #468
automorphism?
homologyAbelianization (H1) = C23×C41 · Schur multiplier (H2) = C2

Aliases

direct productC41×#501

Subgroups

subgroupG/H
center, socle, prime radicalC22×C41C2×D4
commutator, Frattini, cosocleC2×C4C23×C41
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C41 ◁ C42×C41 ◁ G

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 57
transitive degreeμt(G) = 2624

A minimal faithful permutation representation — 3 generators on 57 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41)(42,43,45,47)(44,48,49,46)(51,54)(52,56)(55,57);(42,44,45,49)(43,46,47,48)(51,55)(54,57);(50,51,52,54,53,55,56,57)

Conjugacy classes & character table

656 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC41×#1660, C41×#1661, C41×#1663, …
maximal normal subgroup ofC41×#1660, C41×#1661, C41×#1663, …
maximal quotient ofC41×#1847, C41×#1943, C41×#1944, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2624, 182);