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C41×H(3,ℤ/4) #14579805

order2624
nilpotency class2
isoclinismisoclinic class of H(3,ℤ/4)
automorphism?
homologyAbelianization (H1) = C42×C41 · Schur multiplier (H2) = C2×C4

Aliases

direct productC41×H(3,ℤ/4)

Subgroups

subgroupG/H
centerC4×C41C42
commutatorC4C42×C41
Frattini, cosocleC22×C4C22×C41
socleC2×C41#95
prime radicalC23×C41D4
LCSG ▷ C4 ▷ C1
UCSC1 ◁ C4×C41 ◁ G

94 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)(42,43)(44,50,52,56)(45,48)(46,49)(47,51,54,57)(53,55);(42,44)(43,47,49,54)(45,51)(46,52)(48,50,55,56)(53,57)

Conjugacy classes & character table

902 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC41×#1339, C41×#1340, C2×C41×H(3,ℤ/4), …
maximal normal subgroup ofC41×#1339, C41×#1340, C2×C41×H(3,ℤ/4), …
maximal quotient ofC41×#1274, C41×#1275, C41×#1284, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2624, 18);