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C2×D6141 #37589576

order24564
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD12282
direct productC2×D6141
unambiguous semidirect product(C2×C3)⋊D2047
unambiguous semidirect product(C2×C23)⋊D267
unambiguous semidirect product(C2×C89)⋊D69

Splittings

Fitting, socleC2×C3×C23×C89 ⋊ C2
derived, HallC3×C23×C89 ⋊ C22
HallC23×C89 ⋊ C2×D3
HallC3×C89 ⋊ C2×D23
SylowC89 ⋊ C2×D69
HallC3×C23 ⋊ C2×D89
SylowC23 ⋊ C2×D267
SylowC3 ⋊ C2×D2047

Subgroups

subgroupG/H
center, hypercenterC2D6141
commutator, cosocle, hypocenterC3×C23×C89C22
FrattiniC1G
Fitting, socleC2×C3×C23×C89C2
LCSG ▷ C3×C23×C89
UCSC1 ◁ C2

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 117
transitive degreeμt(G) = 12282

A minimal faithful permutation representation — 2 generators on 117 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)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117);(4,5)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(30,117)(31,116)(32,115)(33,114)(34,113)(35,112)(36,111)(37,110)(38,109)(39,108)(40,107)(41,106)(42,105)(43,104)(44,103)(45,102)(46,101)(47,100)(48,99)(49,98)(50,97)(51,96)(52,95)(53,94)(54,93)(55,92)(56,91)(57,90)(58,89)(59,88)(60,87)(61,86)(62,85)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)

Conjugacy classes & character table

6144 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofD24564
maximal normal subgroup ofD24564
maximal quotient ofDic12282, D24564

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24564, 28);