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Dic1073⋊C3 #27166143

order12876
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productDic1073⋊C3

Splittings

derived, HallC29×C37 ⋊ C3×C4
HallDic1073 ⋊ C3
HallC29×C37⋊C3 ⋊ C4
HallC37⋊C3 ⋊ Dic29
SylowC37 ⋊ C3×Dic29
SylowC29 ⋊ Dic37⋊C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC2C29⋊(D37⋊C3)
commutator, hypocenterC29×C37C3×C4
Fitting, socle, cosocleC2×C29×C37C2×C3
prime radicalC2×C29×C37⋊C3C2
LCSG ▷ C29×C37
UCSC1 ◁ C2

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 70
transitive degreeμt(G) = 4292

A minimal faithful permutation representation — 2 generators on 70 points:

(2,6,7,3,4,8)(5,14,15,9,10,23)(11,26,27,17,18,34)(12,25,21,19,16,13)(20,36,37,29,28,32)(22,24,35,31,30,33)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52)(53,54)(55,56)(57,58)(59,60)(61,62)(63,64)(65,66)(67,68,69,70);(1,2,5,11,18,28,33,34,31,21,7,6,12,20,30,10,16,23,32,37,15,25,14,24,29,19,4,8,13,22,27,35,36,26,17,9,3)(38,39,41,43,45,47,49,51,53,55,57,59,61,63,65,66,64,62,60,58,56,54,52,50,48,46,44,42,40)

Polycyclic presentation:

2,3,29,37,2|a^2=e,c^a=c^28,d^a=d^36,d^b=d^26

Conjugacy classes & character table

444 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#50311168, C3×Dic1073⋊C3, C3⋊(Dic1073⋊C3)
maximal normal subgroup of#50311168, C3×Dic1073⋊C3
maximal quotient of#50311125, C3×Dic1073⋊C3, C3⋊(Dic1073⋊C3)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12876, 12);