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Dic357 #13083609

order1428
isoclinismisoclinic class of D357
automorphismAut(G) order 137088
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C1

Aliases

dicyclicDic357
unambiguous semidirect productC3⋊Dic119
unambiguous semidirect productC7⋊Dic51

Splittings

derived, HallC3×C7×C17 ⋊ C4
HallC7×C17 ⋊ Dic3
HallC3×C17 ⋊ Dic7
HallC3×C7 ⋊ Dic17
SylowC17 ⋊ Dic21
SylowC7 ⋊ Dic51
SylowC3 ⋊ Dic119

Subgroups

subgroupG/H
center, Frattini, hypercenterC2D357
commutator, hypocenterC3×C7×C17C4
Fitting, socle, prime radical, cosocleC2×C3×C7×C17C2
LCSG ▷ C3×C7×C17
UCSC1 ◁ C2

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 31
transitive degreeμt(G) = 1428

A minimal faithful permutation representation — 2 generators on 31 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);(1,3,2,4)(6,7)(9,14)(10,13)(11,12)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)

Polycyclic presentation:

2^2,3,7,17|a^2=b,c^a=c^2,d^a=d^6,e^a=e^16

Conjugacy classes & character table

360 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic1071, C3⋊(ΓL(1,𝔽8)⋊Dic17), C3×Dic357, …
maximal normal subgroup ofC3⋊(ΓL(1,𝔽8)⋊Dic17), C3×Dic357, C11×Dic357, …
maximal quotient ofDic1071, C3×Dic357, C3⋊Dic357, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1428, 11);