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C2×D7503 #44473154

order30012
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD15006
direct productC2×D7503
unambiguous semidirect product(C2×C3)⋊D2501
unambiguous semidirect product(C2×C41)⋊D183
unambiguous semidirect product(C2×C61)⋊D123

Splittings

Fitting, socleC2×C3×C41×C61 ⋊ C2
derived, HallC3×C41×C61 ⋊ C22
HallC41×C61 ⋊ C2×D3
HallC3×C61 ⋊ C2×D41
HallC3×C41 ⋊ C2×D61
SylowC61 ⋊ C2×D123
SylowC41 ⋊ C2×D183
SylowC3 ⋊ C2×D2501

Subgroups

subgroupG/H
center, hypercenterC2D7503
commutator, cosocle, hypocenterC3×C41×C61C22
FrattiniC1G
Fitting, socleC2×C3×C41×C61C2
LCSG ▷ C3×C41×C61
UCSC1 ◁ C2

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 107
transitive degreeμt(G) = 15006

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

Conjugacy classes & character table

7506 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofD30012
maximal normal subgroup ofD30012
maximal quotient ofDic15006, D30012

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30012, 52);