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Cayley diagram

C52⋊C3 #658

order75
isoclinismisoclinic class of C52⋊C3
automorphismAut(G) = AΓL(1,𝔽25) (order 1200) · Inn(G) = G · Out(G) = M16
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C5

Aliases

unambiguous semidirect productC52⋊C3

Splittings

Fitting, socle, derived, SylowC52 ⋊ C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC52C3
LCSG ▷ C52
UCSC1

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 15

A minimal faithful permutation representation — 2 generators on 15 points:

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

Polycyclic presentation:

3,5^2|b^a=bc^2,c^a=bc^3

Conjugacy classes & character table

C1C2C3C4C5C6C7C8C9C10C11
size1333333332525
order15555555533
χ111111111111
χ2111111111−1+i32−1−i32
χ3111111111−1−i32−1+i32
χ43−2ζ−ζ8−2ζ11−ζ131−521−52−ζ−2ζ2−2ζ7−ζ111+521+52−ζ2−2ζ4−ζ7−2ζ14−ζ4−2ζ8−2ζ13−ζ1400
χ531−52−ζ−2ζ2−2ζ7−ζ11−ζ4−2ζ8−2ζ13−ζ141+52−2ζ−ζ8−2ζ11−ζ13−ζ2−2ζ4−ζ7−2ζ141−521+5200
χ631−52−ζ4−2ζ8−2ζ13−ζ14−ζ−2ζ2−2ζ7−ζ111+52−ζ2−2ζ4−ζ7−2ζ14−2ζ−ζ8−2ζ11−ζ131−521+5200
χ73−ζ−2ζ2−2ζ7−ζ111+521+52−ζ2−2ζ4−ζ7−2ζ141−521−52−ζ4−2ζ8−2ζ13−ζ14−2ζ−ζ8−2ζ11−ζ1300
χ831+52−2ζ−ζ8−2ζ11−ζ13−ζ2−2ζ4−ζ7−2ζ141−52−ζ4−2ζ8−2ζ13−ζ14−ζ−2ζ2−2ζ7−ζ111+521−5200
χ931+52−ζ2−2ζ4−ζ7−2ζ14−2ζ−ζ8−2ζ11−ζ131−52−ζ−2ζ2−2ζ7−ζ11−ζ4−2ζ8−2ζ13−ζ141+521−5200
χ103−ζ2−2ζ4−ζ7−2ζ141−521−52−ζ4−2ζ8−2ζ13−ζ141+521+52−2ζ−ζ8−2ζ11−ζ13−ζ−2ζ2−2ζ7−ζ1100
χ113−ζ4−2ζ8−2ζ13−ζ141+521+52−2ζ−ζ8−2ζ11−ζ131−521−52−ζ−2ζ2−2ζ7−ζ11−ζ2−2ζ4−ζ7−2ζ1400

Here ζ is a primitive root of unity of order 15 (ζ15 = 1); i = ζ3.

Relationships

derived subgroup of#3874, #64470, #64480, …
maximal subgroup of#3874, (C5⋊D5)⋊C3, C2×C52⋊C3, …
maximal normal subgroup of#3874, (C5⋊D5)⋊C3, C2×C52⋊C3, …
maximal quotient ofC2×C52⋊C3, C52⋊C9, C3×C52⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(75, 2);