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C4⋊C4×#11719921 #27305505

order13568
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4

Aliases

direct productC4⋊C4×#11719921

Splittings

FittingC2×C4×C4⋊C4×C53 ⋊ C2
SylowC53 ⋊ C22⋊C4×C4⋊C4

Subgroups

subgroupG/H
center, FrattiniC24C23×D53
commutatorC22×C53C22×C42
FittingC2×C4×C4⋊C4×C53C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenterC2×C4×C4⋊C4D53
hypocenterC53C22⋊C4×C4⋊C4
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4

2574 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 69
transitive degreeμt(G) = 6784

A minimal faithful permutation representation — 4 generators on 69 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(58,59,60,61)(67,69);(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)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(66,67)(68,69);(62,63,64,65);(54,55,56,57)(63,65)

Conjugacy classes & character table

2180 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 90981);