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Cayley diagram

C22×D53 #88334

order424
isoclinismisoclinic class of D53
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23

Aliases

direct productC22×D53
unambiguous semidirect productC53⋊C23
unambiguous semidirect product(C2×C53)⋊C22

Splittings

Fitting, socleC22×C53 ⋊ C2
derived, SylowC53 ⋊ C23

Subgroups

subgroupG/H
center, hypercenterC22D53
commutator, cosocle, hypocenterC53C23
FrattiniC1G
Fitting, socleC22×C53C2
LCSG ▷ C53
UCSC1 ◁ C22

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 57
transitive degreeμt(G) = 212

A minimal faithful permutation representation — 3 generators on 57 points:

(1,2)(3,4)(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)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57);(3,4)(5,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7);(1,2)

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11719921, C22⋊#6157, C2×C4×D53, …
maximal normal subgroup of#11719921, C22⋊#6157, C2×C4×D53, …
maximal quotient ofC2×Dic106, C2×C4×D53, C2×D212, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(424, 13);