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C2×D13725 #72290752

order54900
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD27450
direct productC2×D13725

Splittings

FittingC2×C9×C25×C61 ⋊ C2
derived, HallC9×C25×C61 ⋊ C22
HallC25×C61 ⋊ C2×D9
HallC9×C61 ⋊ C2×D25
HallC9×C25 ⋊ C2×D61
SylowC61 ⋊ C2×D225
SylowC25 ⋊ C2×D549
SylowC9 ⋊ C2×D1525

Subgroups

subgroupG/H
center, hypercenterC2D13725
commutator, cosocle, hypocenterC9×C25×C61C22
FrattiniC3×C5C2×D915
FittingC2×C9×C25×C61C2
socleC2×C3×C5×C61D15
LCSG ▷ C9×C25×C61
UCSC1 ◁ C2

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 97
transitive degreeμt(G) = 27450

A minimal faithful permutation representation — 2 generators on 97 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);(4,11)(5,10)(6,9)(7,8)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(38,97)(39,96)(40,95)(41,94)(42,93)(43,92)(44,91)(45,90)(46,89)(47,88)(48,87)(49,86)(50,85)(51,84)(52,83)(53,82)(54,81)(55,80)(56,79)(57,78)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)

Conjugacy classes & character table

13728 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((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), (4,11)(5,10)(6,9)(7,8)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(38,97)(39,96)(40,95)(41,94)(42,93)(43,92)(44,91)(45,90)(46,89)(47,88)(48,87)(49,86)(50,85)(51,84)(52,83)(53,82)(54,81)(55,80)(56,79)(57,78)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68));