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C2×D7777 #44527425

order31108
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD15554
direct productC2×D7777
unambiguous semidirect product(C2×C7)⋊D1111
unambiguous semidirect product(C2×C11)⋊D707
unambiguous semidirect product(C2×C101)⋊D77

Splittings

Fitting, socleC2×C7×C11×C101 ⋊ C2
derived, HallC7×C11×C101 ⋊ C22
HallC11×C101 ⋊ C2×D7
HallC7×C101 ⋊ C2×D11
SylowC101 ⋊ C2×D77
HallC7×C11 ⋊ C2×D101
SylowC11 ⋊ C2×D707
SylowC7 ⋊ C2×D1111

Subgroups

subgroupG/H
center, hypercenterC2D7777
commutator, cosocle, hypocenterC7×C11×C101C22
FrattiniC1G
Fitting, socleC2×C7×C11×C101C2
LCSG ▷ C7×C11×C101
UCSC1 ◁ C2

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 121
transitive degreeμt(G) = 15554

A minimal faithful permutation representation — 2 generators on 121 points:

(1,2)(3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121);(4,9)(5,8)(6,7)(11,20)(12,19)(13,18)(14,17)(15,16)(22,121)(23,120)(24,119)(25,118)(26,117)(27,116)(28,115)(29,114)(30,113)(31,112)(32,111)(33,110)(34,109)(35,108)(36,107)(37,106)(38,105)(39,104)(40,103)(41,102)(42,101)(43,100)(44,99)(45,98)(46,97)(47,96)(48,95)(49,94)(50,93)(51,92)(52,91)(53,90)(54,89)(55,88)(56,87)(57,86)(58,85)(59,84)(60,83)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)

Conjugacy classes & character table

7780 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofD31108
maximal normal subgroup ofD31108
maximal quotient ofDic15554, D31108

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31108, 27);