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Dic301 #11925839

order1204
isoclinismisoclinic class of D301
automorphismAut(G) order 151704
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C1

Aliases

dicyclicDic301
unambiguous semidirect productC7⋊Dic43
unambiguous semidirect productC43⋊Dic7

Splittings

derived, HallC7×C43 ⋊ C4
SylowC43 ⋊ Dic7
SylowC7 ⋊ Dic43

Subgroups

subgroupG/H
center, Frattini, hypercenterC2D301
commutator, hypocenterC7×C43C4
Fitting, socle, prime radical, cosocleC2×C7×C43C2
LCSG ▷ C7×C43
UCSC1 ◁ C2

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 54
transitive degreeμt(G) = 1204

A minimal faithful permutation representation — 2 generators on 54 points:

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

Polycyclic presentation:

2^2,7,43|a^2=b,c^a=c^6,d^a=d^42

Conjugacy classes & character table

304 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C43⋊C3)⋊Dic7, ΓL(1,𝔽8)⋊Dic43, #16813406, …
maximal normal subgroup of(C43⋊C3)⋊Dic7, ΓL(1,𝔽8)⋊Dic43, #16813406, …
maximal quotient ofC3×Dic301, Dic903, C7⋊(C43⋊#157), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1204, 5);