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D300 #10597046

order600
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD300
unambiguous semidirect productC4⋊D75

Splittings

FittingC3×C4×C25 ⋊ C2
HallC3×C25 ⋊ D4
SylowC25 ⋊ D12
SylowC3 ⋊ D100

Subgroups

subgroupG/H
centerC2C2×D75
commutator, cosocleC2×C3×C25C22
FrattiniC2×C5C2×D15
FittingC3×C4×C25C2
socleC2×C3×C5C2×D5
hypercenterC4D75
hypocenterC3×C25D4
LCSG ▷ C2×C3×C25 ▷ C3×C25
UCSC1 ◁ C2 ◁ C4

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 32
transitive degreeμt(G) = 300

A minimal faithful permutation representation — 2 generators on 32 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);(2,4)(6,7)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)

Polycyclic presentation:

2^3,3,5^2|b^2=c,e^5=f,b^a=bc,d^a=d^2,e^a=e^4f^4,f^a=f^4

Conjugacy classes & character table

153 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11924802, #11924803, #11924808, …
maximal normal subgroup of#11924802, #11924803, #11924808, …
maximal quotient ofDic300, #11924858, D600, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(600, 27);