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C2×D6097 #37567236

order24388
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD12194
direct productC2×D6097
unambiguous semidirect product(C2×C7)⋊D871
unambiguous semidirect product(C2×C13)⋊D469
unambiguous semidirect product(C2×C67)⋊D91

Splittings

Fitting, socleC2×C7×C13×C67 ⋊ C2
derived, HallC7×C13×C67 ⋊ C22
HallC13×C67 ⋊ C2×D7
HallC7×C67 ⋊ C2×D13
HallC7×C13 ⋊ C2×D67
SylowC67 ⋊ C2×D91
SylowC13 ⋊ C2×D469
SylowC7 ⋊ C2×D871

Subgroups

subgroupG/H
center, hypercenterC2D6097
commutator, cosocle, hypocenterC7×C13×C67C22
FrattiniC1G
Fitting, socleC2×C7×C13×C67C2
LCSG ▷ C7×C13×C67
UCSC1 ◁ C2

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 89
transitive degreeμt(G) = 12194

A minimal faithful permutation representation — 2 generators on 89 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);(4,9)(5,8)(6,7)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(24,89)(25,88)(26,87)(27,86)(28,85)(29,84)(30,83)(31,82)(32,81)(33,80)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)

Conjugacy classes & character table

6100 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofD24388
maximal normal subgroup ofD24388
maximal quotient ofDic12194, D24388

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24388, 27);