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C7×C3⋊#1042 #14523653

order2100
isoclinismisoclinic class of C3⋊#1042
automorphism?
homologyAbelianization (H1) = C4×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×C3⋊#1042
unambiguous semidirect productC3⋊(C7×#1042)

Splittings

Fitting, socle, HallC3×C52×C7 ⋊ C4
derived, HallC3×C52 ⋊ C4×C7
HallC52×C7 ⋊ Dic3
SylowC52 ⋊ C7×Dic3
HallC3×C7 ⋊ #1042
SylowC3 ⋊ C7×#1042

Subgroups

subgroupG/H
center, hypercenterC7C3⋊#1042
commutator, hypocenterC3×C52C4×C7
FrattiniC1G
Fitting, socleC3×C52×C7C4
prime radicalC3×C5×C7×D5C2
cosocleC3×C5×D5C2×C7
LCSG ▷ C3×C52
UCSC1 ◁ C7

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 420

A minimal faithful permutation representation — 2 generators on 20 points:

(1,2,3,4,5,6,7)(9,10)(12,13)(14,15)(17,18,19,20);(8,9,10)(11,12,14,15,13)(16,17,18,20,19)

Conjugacy classes & character table

231 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC49×C3⋊#1042, C72×C3⋊#1042, C7×C3⋊(C7⋊#1042), …
maximal normal subgroup ofC49×C3⋊#1042, C72×C3⋊#1042, C7×C11×C3⋊#1042, …
maximal quotient ofC49×C3⋊#1042, C72×C3⋊#1042, C7×C3⋊(C7⋊#1042), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2100, 133);