Finite Group Databasebeta

← Prev (#10597035) Next (#10597037) →

C3×D100 #10597036

order600
isoclinismisoclinic class of Dic50
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productC3×D100
unambiguous semidirect productC4⋊(C3×D25)

Splittings

FittingC3×C4×C25 ⋊ C2
HallC3×C25 ⋊ D4
SylowC25 ⋊ C3×D4

Subgroups

subgroupG/H
centerC2×C3C2×D25
commutator, cosocleC2×C25C22×C3
FrattiniC2×C5C2×C3×D5
FittingC3×C4×C25C2
socleC2×C3×C5C2×D5
hypercenterC3×C4D25
hypocenterC25C3×D4
LCSG ▷ C2×C25 ▷ C25
UCSC1 ◁ C2×C3 ◁ C3×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);(5,7)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)

Conjugacy classes & character table

159 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11924801, #11924803, #11924807, …
maximal normal subgroup of#11924801, #11924803, #11924807, …
maximal quotient ofC3×Dic100, C3×#87713, C3×D200, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(600, 17);