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C2×#72221874 #72253997

order8748
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C34

Aliases

direct productC2×#72221874

Splittings

FittingC2×#14555436 ⋊ C2
Sylow#14555436 ⋊ C22

Subgroups

subgroupG/H
center, socleC2×C3C3×C3⋊(C3⋊((C3⋊D3)⋊C3))
commutator, cosocle, hypocenterH(5,𝔽3)C22×C32
FrattiniC32C2×C32×C3⋊(C3⋊D3)
FittingC2×#14555436C2
hypercenterC2×C32C3⋊(C3⋊((C3⋊D3)⋊C3))
LCSG ▷ H(5,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C32

8690 conjugacy classes of subgroups; only the 444 normal subgroups are shown.

normal subgroups only

This group has 8690 conjugacy classes of subgroups — too many to draw. Only its 444 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 83
transitive degreeμt(G) = 162

A minimal faithful permutation representation — 4 generators on 83 points:

(2,8,11)(3,30,17,12,13,37)(4,35,18,15,16,36)(7,19,22)(9,49,23,25,26,46)(10,54,24,28,29,48)(14,57,38,33,32,59)(20,41)(21,44)(27,74,47,52,51,71)(31,34)(39,61)(40,63)(42,66)(43,65)(45,69)(50,72,73,53,70,75)(55,58)(56,60)(62,77)(64,78,79,67,76,81)(68,80);(1,2,7)(3,9,20)(4,24,40)(5,11,19)(6,8,22)(10,45,18)(12,25,41)(13,23,43)(14,47,62)(15,54,69)(16,29,21)(17,26,39)(27,68,38)(28,63,35)(30,46,65)(31,70,76)(32,51,42)(33,74,80)(34,75,81)(36,48,44)(37,49,61)(50,79,56)(52,77,57)(53,78,58)(55,73,64)(59,71,66)(60,72,67)(82,83);(1,4,15)(2,10,28)(3,14,34)(5,18,36)(6,16,35)(7,21,44)(8,24,48)(9,27,53)(11,29,54)(12,31,33)(13,32,58)(17,38,60)(19,40,63)(20,42,67)(22,45,69)(23,47,72)(25,50,52)(26,51,75)(30,55,57)(37,56,59)(39,62,78)(41,64,66)(43,68,81)(46,70,71)(49,73,74)(61,76,77)(65,79,80);(1,3,12)(2,9,25)(4,14,31)(5,17,37)(6,13,30)(7,20,41)(8,23,46)(10,27,50)(11,26,49)(15,34,33)(16,32,55)(18,38,56)(19,39,61)(21,42,64)(22,43,65)(24,47,70)(28,53,52)(29,51,73)(35,58,57)(36,60,59)(40,62,76)(44,67,66)(45,68,79)(48,72,71)(54,75,74)(63,78,77)(69,81,80)

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC2×#72221884
maximal normal subgroup ofC2×#72221884

GAP

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

gap> G := Group((2,8,11)(3,30,17,12,13,37)(4,35,18,15,16,36)(7,19,22)(9,49,23,25,26,46)(10,54,24,28,29,48)(14,57,38,33,32,59)(20,41)(21,44)(27,74,47,52,51,71)(31,34)(39,61)(40,63)(42,66)(43,65)(45,69)(50,72,73,53,70,75)(55,58)(56,60)(62,77)(64,78,79,67,76,81)(68,80), (1,2,7)(3,9,20)(4,24,40)(5,11,19)(6,8,22)(10,45,18)(12,25,41)(13,23,43)(14,47,62)(15,54,69)(16,29,21)(17,26,39)(27,68,38)(28,63,35)(30,46,65)(31,70,76)(32,51,42)(33,74,80)(34,75,81)(36,48,44)(37,49,61)(50,79,56)(52,77,57)(53,78,58)(55,73,64)(59,71,66)(60,72,67)(82,83), (1,4,15)(2,10,28)(3,14,34)(5,18,36)(6,16,35)(7,21,44)(8,24,48)(9,27,53)(11,29,54)(12,31,33)(13,32,58)(17,38,60)(19,40,63)(20,42,67)(22,45,69)(23,47,72)(25,50,52)(26,51,75)(30,55,57)(37,56,59)(39,62,78)(41,64,66)(43,68,81)(46,70,71)(49,73,74)(61,76,77)(65,79,80), (1,3,12)(2,9,25)(4,14,31)(5,17,37)(6,13,30)(7,20,41)(8,23,46)(10,27,50)(11,26,49)(15,34,33)(16,32,55)(18,38,56)(19,39,61)(21,42,64)(22,43,65)(24,47,70)(28,53,52)(29,51,73)(35,58,57)(36,60,59)(40,62,76)(44,67,66)(45,68,79)(48,72,71)(54,75,74)(63,78,77)(69,81,80));