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D4×C52⋊C3 #10597131

order600
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C5

Aliases

direct productD4×C52⋊C3
unambiguous semidirect productC4⋊(C2×C52⋊C3)
unambiguous semidirect product(C52×D4)⋊C3

Splittings

Fitting, HallC52×D4 ⋊ C3
SylowC52 ⋊ C3×D4

Subgroups

subgroupG/H
center, FrattiniC2C22×C52⋊C3
commutator, socle, cosocleC2×C52C22×C3
FittingC52×D4C3
hypercenterD4C52⋊C3
hypocenterC52C3×D4
LCSG ▷ C2×C52 ▷ C52
UCSC1 ◁ C2 ◁ D4

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 60

A minimal faithful permutation representation — 2 generators on 19 points:

(1,2,3,4)(5,6,8)(7,11,13)(9,14,17)(10,16,19)(12,15,18);(2,4)(5,7,12,17,19)(6,11,15,9,10)(8,14,13,16,18)

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11925029, #11925030, #87809⋊C3, …
maximal normal subgroup of#11925029, #11925030, #87809⋊C3, …
maximal quotient ofC22⋊C4×C52⋊C3, C4⋊C4×C52⋊C3, D8×C52⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(600, 112);