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C41×#328 #14579796

order2624
nilpotency class3
isoclinismisoclinic class of #327
automorphism?
homologyAbelianization (H1) = C2×C4×C41 · Schur multiplier (H2) = C2

Aliases

direct productC41×#328

Subgroups

subgroupG/H
center, socleC22×C41C22⋊C4
commutatorC2×C4C2×C4×C41
Frattini, cosocleC22×C4C22×C41
prime radicalC23×C41D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C41 ◁ C22×C4×C41 ◁ G

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 57
transitive degreeμt(G) = 656

A minimal faithful permutation representation — 2 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)(43,47,48,56)(44,51)(49,57,50,52)(53,55);(42,43,45,49)(44,52,51,56)(46,50,54,48)(47,55,57,53)

Conjugacy classes & character table

779 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×C41×#328, C41×#1500, C41×#1501, …
maximal normal subgroup ofC2×C41×#328, C41×#1500, C41×#1501, …
maximal quotient ofC41×#1271, C41×#1272, C41×#1294, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2624, 9);