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C2×C7×C3⋊D321 #41004160

order26964
automorphism?
homologyAbelianization (H1) = C22×C7 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×C7×C3⋊D321
unambiguous semidirect product(C2×C3)⋊(C7×D321)
unambiguous semidirect product(C2×C107)⋊(C7×C3⋊D3)

Splittings

Fitting, socleC2×C32×C7×C107 ⋊ C2
derived, HallC32×C107 ⋊ C22×C7
HallC32×C7×C107 ⋊ C22
HallC7×C107 ⋊ C2×C3⋊D3
SylowC107 ⋊ C2×C7×C3⋊D3
HallC32×C7 ⋊ C2×D107
SylowC32 ⋊ C2×C7×D107

Subgroups

subgroupG/H
center, hypercenterC2×C7C3⋊D321
commutator, cosocle, hypocenterC32×C107C22×C7
FrattiniC1G
Fitting, socleC2×C32×C7×C107C2
LCSG ▷ C32×C107
UCSC1 ◁ C2×C7

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 122
transitive degreeμt(G) = 13482

A minimal faithful permutation representation — 3 generators on 122 points:

(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)(116,117)(119,120);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,100,102,104,106,107,105,103,101,99,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(118,119,120)(121,122);(115,116,117)

Conjugacy classes & character table

6762 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26964, 109);