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D5×#3585⋊C3 #72277417

order3840
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C26

Aliases

direct productD5×#3585⋊C3

Splittings

FittingC5×#3585 ⋊ C2×C3
HallD5×#3585 ⋊ C3
SylowC5 ⋊ C2×#3585⋊C3

Subgroups

subgroupG/H
centerC2C2×D4×D5×A4
commutator, cosocleC5×#41C24×C3
FrattiniC4C23×D5×A4
FittingC5×#3585C2×C3
socleC2×C5C22×D4×A4
hypercenter#137D5×A4
hypocenterC5×Q8C22×C3×D4
LCSG ▷ C5×#41 ▷ C5×Q8
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

4536 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 53
transitive degreeμt(G) = 240

A minimal faithful permutation representation — 4 generators on 53 points:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

GAP

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

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