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C2×#495574 #72308339

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4

Aliases

direct productC2×#495574

Subgroups

subgroupG/H
center, socleC22#21408
commutatorC2×C8C24×C4
Frattini, cosocleC22×C8C25
prime radicalC2×#3060C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#2875 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

5049 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 66
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 5 generators on 66 points:

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

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#72249778, C2×#72307114, C2×#72249754
maximal normal subgroup ofC2×#72249778, C2×#72307114, C2×#72249754

GAP

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

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